c_{p,n} &= \Re(c_{p,n}) = \Re\left(\frac 1\pi \int_\R g_n(x) \cdot \frac{n\left(\left(x^2 - p^2 - \frac 1{n^2}\right) - i\frac{2x}{n}\right)\left(\frac 1n + ix\right)\left(1+i\frac xn\right)}{\left(x^2 - p^2 - \frac 1{n^2}\right)^2 + 4x^2\frac{1}{n^2}} |F(x)|^2 \,dx\right)
\\&= \frac 1\pi \int_\R g_n(x) \cdot \frac{\Re\left(n\left(\left(x^2 - p^2 - \frac 1{n^2}\right) - i\frac{2x}{n}\right)\left(\frac 1n + ix\right)\left(1+i\frac xn\right)\right)}{\left(x^2 - p^2 - \frac 1{n^2}\right)^2 + 4x^2\frac{1}{n^2}} |F(x)|^2 \,dx
\\&= \frac 1\pi \int_\R g_n(x) \cdot \frac{\left(x^2 - p^2 - \frac 1{n^2}\right)(1-x^2) + 2x^2\left(\frac 1{n^2}  + 1\right)}{\left(x^2 - p^2 - \frac 1{n^2}\right)^2 + 4x^2\frac{1}{n^2}} |F(x)|^2 \,dx,
\end{align*}

so defining the function

\begin{align*}
d_{p,n}(x) \coloneqq g_n(x) \cdot \frac{\left(x^2-p^2-\frac 1{n^2}\right)\left(1-x^2\right)+2x^2\left(\frac 1{n^2} + 1\right)}{\left(x^2-p^2-\frac 1{n^2}\right)^2+4x^2 \frac 1{n^2}}, \quad x \in \R,
\end{align*}

we have

\[c_{p,n} = \frac 1\pi \int_\R d_{p,n}(x) |F(x)|^2 \,dx.\]

Then defining

\begin{align*}
f_{p,n}(x) \coloneqq d_{p,n}(x) + g_n(x) \cdot \left(\chi_{\left(-1,1\right)}(x) \cdot \frac 1{p^2} + \chi_{\left(-\infty,-1\right) \cup \left(1,\infty\right)}(x)\right), \quad x \in \R,
\end{align*}

we have

\[\frac 1\pi \int_\R f_{p,n}(x) |F(x)|^2 \,dx = c_{p,n} + \frac 1\pi a_n \cdot \frac 1{p^2} + \frac 1\pi b_n \quad \forall n \in \N.\]

This, by thm:bigIntegralApproximation, yields

\begin{align*}
|F(p)|^2 &= \frac{|F(p)|^2 + |F(-p)|^2}2 = \lim_{n \to \infty} \frac 1\pi \int_\R f_{p,n}(x) |F(x)|^2 \,dx
\\&= \lim_{n \to \infty} \left(c_{p,n} + \frac 1\pi a_n \cdot \frac 1{p^2} + \frac 1\pi b_n\right)
\end{align*}

for almost every \(p \in \R\). Since \(a_n,b_n,c_{p,n} \geq 0\) this implies that the sequences \((a_n)_{n \in \N}\), \((b_n)_{n \in \N}\) and \((c_{p,n})_{n \in \N}\) all are bounded. Therefore there exists a subsequence \((n_k)_{k \in \N}\) such that \((a_{n_k})_{k \in \N}\) and \((b_{n_k})_{k \in \N}\) are convergent. Setting

\[A \coloneqq \lim_{k \to \infty} a_{n_k} \qquad \text{and} \qquad B \coloneqq \lim_{k \to \infty} b_{n_k},\]

we then get

\[|F(p)|^2 = \lim_{k \to \infty} \left(c_{p,{n_k}} + \frac 1\pi a_{n_k} \cdot \frac 1{p^2} + \frac 1\pi b_{n_k} \cdot 1\right) = \left(\lim_{k \to \infty} c_{p,{n_k}}\right) + \frac 1\pi A \cdot \frac 1{p^2} + \frac 1\pi B\]

for almost every \(p \in \R\). Therefore, for almost every \(p \in \R\), the sequence \((c_{p,n_k})_{k \in \N}\) is convergent. In this case, we set

\[C_p \coloneqq \lim_{k \to \infty} c_{p,{n_k}}.\]

Now, by cor:thetaHHasMeasure, there exists a Borel measure \(\mu \in \mathcal{BM}\left(\R_+\right)\) such that

\[\braket*{f}{\theta_h g} = \int_{\R_+} \overline{f(i\lambda)} g(i\lambda) \,d\mu(\lambda) \quad \forall f,g \in H^2(\C_+).\]

This, using that

\[\Lambda(i\lambda) = \frac{ii\lambda}{(i\lambda+i)^2} = \frac \lambda{(1+\lambda)^2}, \quad \lambda \in \R_+\]

and that \(\sqrt{F_{p,n}}(i\lambda) > 0\), yields

\begin{align*}
c_{p,n} &=\braket*{\sqrt{F_{p,n}}\Lambda F}{\theta_h\sqrt{F_{p,n}}\Lambda F}
\\&= \int_{\R_+} \overline{\sqrt{F_{p,n}}(i\lambda)\Lambda(i\lambda)F(i\lambda)} \cdot \sqrt{F_{p,n}}(i\lambda)\Lambda(i\lambda)F(i\lambda) \,d\mu(\lambda)
\\&= \int_{\R_+} F_{p,n}(i\lambda)\Lambda(i\lambda)^2\left|F(i\lambda)\right|^2 \,d\mu(\lambda)
\\&= \int_{\R_+} \frac 1\pi \cdot \frac{(1+\lambda)^4}{\left(p^2+\left(\lambda+\frac 1n\right)^2\right)\left(\frac 1n+\lambda\right)\left(1+\frac \lambda n\right)}\left(\frac \lambda{(1+\lambda)^2}\right)^2\left|F(i\lambda)\right|^2 \,d\mu(\lambda)
\\&= \frac 1\pi \int_{\R_+} \frac{\lambda^2}{\left(p^2+\left(\lambda+\frac 1n\right)^2\right)\left(\frac 1n+\lambda\right)\left(1+\frac \lambda n\right)}\left|F(i\lambda)\right|^2 \,d\mu(\lambda).
\end{align*}

Then, by the Monotone Convergence Theorem, we get

\begin{align*}
C_p = \lim_{k \to \infty} c_{p,n_k} &= \lim_{k \to \infty} \frac 1\pi \int_{\R_+} \frac{\lambda^2}{\left(p^2+\left(\lambda+\frac 1{n_k}\right)^2\right)\left(\frac 1{n_k}+\lambda\right)\left(1+\frac \lambda {n_k}\right)}\left|F(i\lambda)\right|^2 \,d\mu(\lambda)
\\&= \frac 1\pi \int_{\R_+} \frac{\lambda^2}{\left(p^2+\lambda^2\right) \cdot \lambda \cdot 1}\left|F(i\lambda)\right|^2 \,d\mu(\lambda)
\\&= \frac 1\pi \int_{\R_+} \frac{1+\lambda^2}{p^2+\lambda^2}\left|F(i\lambda)\right|^2 \frac{\lambda}{1+\lambda^2}\,d\mu(\lambda).
\end{align*}

This in particular implies that the measure \(\tilde \nu\) on \(\R_+\) defined by

\[d\tilde \nu(\lambda) = \left|F(i\lambda)\right|^2 \frac{\lambda}{1+\lambda^2}\,d\mu(\lambda)\]

is finite. We then get a finite measure \(\nu\) on \([0,\infty]\) by

\[\nu\big|_{\R_+} = \tilde \nu, \quad \nu(\{0\}) = A \quad \text{and} \quad \nu(\{\infty\}) = B.\]

Finally, for almost every \(p \in \R\), we have

\begin{align*}
|F(p)|^2 &= C_p + \frac 1\pi A \cdot \frac 1{p^2} + \frac 1\pi B
\\&= \frac 1\pi \int_{\R_+} \frac{1+\lambda^2}{p^2+\lambda^2}\,d\nu(\lambda) + \frac 1\pi \nu(\{0\}) \cdot \frac 1{p^2} + \frac 1\pi \nu(\{\infty\}) \cdot 1
\\&= \frac 1\pi \int_{[0,\infty]} \frac{1+\lambda^2}{p^2+\lambda^2}\,d\nu(\lambda) = \Psi_\nu(p). \qedhere
\end{align*}

Let \(h \in L^\infty(\R,\T)^\flat\).
Then the following are equivalent:

* \(\left(L^2\left(\R,\C\right),H^2\left(\C_+\right),\theta_h\right)\) is a maximal RPHS.

* \(h = h_{\nu}\) for some measure \(\nu \in \mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\}\).

That (b) implies (a) is precisely the statement of cor:hNuMaxPos. We now show that also the converse is true. So let \(\left(L^2\left(\R,\C\right),H^2\left(\C_+\right),\theta_h\right)\) be a maximal RPHS. Then, by cor:kerNonTrivOuterFixExtended, there exists an outer function \(F \in \Out^2_\Lambda(\C_+)\) such that

\[h = \frac F{RF}.\]

By prop:FixedPointFromConstruction this implies that there exists a measure \(\nu \in \mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\}\) and \(C \in \T\) such that

\[F = C \cdot F_\nu.\]

We get

\[h = \frac F{RF} = \frac{C \cdot F_\nu}{C \cdot RF_\nu} = \frac{F_\nu}{RF_\nu} = h_\nu. \qedhere\]

§.§ Uniqueness of the measure

In thm:GeneralizedMaxPosForm we have seen that, for every function \(h \in L^\infty(\R,\T)^\flat\) for which the triple \(\left(L^2\left(\R,\C\right),H^2\left(\C_+\right),\theta_h\right)\) is a maximal RPHS, there exists measure \(\nu \in \mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\}\) such that \(h = h_{\nu}\). In this subsection we want to discuss to which extent this measure is unique. For this we first notice that, for a measure \(\nu \in \mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\}\) and \(c \in \R_+\), by lem:OuterHomo, we have

\[F_{c \cdot \nu} = \Out\left(\sqrt{\Psi_{c \cdot \nu}}\right) = \Out\left(\sqrt{c} \cdot \sqrt{\Psi_\nu}\right) = \sqrt{c} \cdot \Out\left(\sqrt{\Psi_\nu}\right) = \sqrt{c} \cdot F_\nu\]

and therefore

\[h_{c \cdot \nu} = \frac{F_{c \cdot \nu}}{RF_{c \cdot \nu}} = \frac{\sqrt{c} \cdot F_\nu}{\sqrt{c} \cdot RF_\nu} = \frac{F_\nu}{RF_\nu} = h_\nu.\]

This shows that the map

\[\mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\} \ni \nu \mapsto h_\nu\]

is not injective. On the other hand, defining the equivalence relation

\[\nu \gls*{ZZZtilde} \nu' \quad :\Leftrightarrow \quad (\exists c \in \R_+)\, \nu = c \cdot \nu'\]

and denoting the corresponding equivalence classes by \([\nu]\), it shows that the map

\[\left(\mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\}\right) \slash \sim \,\ni [\nu] \mapsto h_\nu\]

is well-defined. We want to show that this map is injective. For this we need the following proposition:

For \(z \in \C_r\) we set

\begin{equation*}
q_z: [0,\infty] \to \C, \quad \lambda \mapsto \frac{\lambda+1}{\lambda+z}.
\end{equation*}

\[\overline{\spann_\C \left\lbrace q_z: z \in \C_r\right\rbrace} = C([0,\infty],\C) \quad \text{and} \quad \overline{\spann_\R \left\lbrace q_z: z \in \R_+\right\rbrace} = C([0,\infty],\R)\]

where the closure is taken with respect to the supremum norm on \([0,\infty]\).

We set

\[A \coloneqq \overline{\spann_\C \left\lbrace q_z: z \in \C_r\right\rbrace}\subeq C([0,\infty],\C) \quad \text{and} \quad B \coloneqq \overline{\spann_\R \left\lbrace q_z: z \in \R_+\right\rbrace}\subeq C([0,\infty],\R).\]

We now show that \(A\) and \(B\) are algebras. For \(z \in \C_r\) one has

\[q_z(\lambda) = \frac{\lambda+1}{\lambda+z} = \frac{1-z}{\lambda+z} + 1.\]

Further, for \(z,w \in \C_r\), we get

\begin{align*}
(1-z)(1-w) \cdot \frac{w-z}{(\lambda + z)(\lambda + w)} &= (1-z)(1-w) \cdot \left(\frac{1}{\lambda+z} - \frac{1}{\lambda+w}\right)
\\&= (1-w) \cdot \frac{1-z}{\lambda+z} - (1-z) \cdot \frac{1-w}{\lambda+w}
\\&= (1-w) \cdot (q_z(\lambda)-1) - (1-z) \cdot (q_w(\lambda)-1).
\end{align*}

If \(z \neq w\) this yields

\begin{align*}
q_z(\lambda) \cdot q_w(\lambda) &= \left(\frac{1-z}{\lambda+z} + 1\right) \cdot \left(\frac{1-w}{\lambda+w} + 1\right)
\\&= \frac{(1-z)(1-w)}{(\lambda+z)(\lambda+w)} + \frac{1-z}{\lambda+z} + \frac{1-w}{\lambda+w} + 1
\\&= \frac{1}{w-z} \cdot \left((1-w) \cdot (q_z(\lambda)-1) - (1-z) \cdot (q_w(\lambda)-1)\right) + q_z(\lambda) + q_w(\lambda) - 1
\\&= \frac{1-z}{w-z} q_z(\lambda) + \frac{1-w}{z-w} q_w(\lambda).
\end{align*}

Therefore, for \(z,w \in \C_r\) with \(z \neq w\) one has \(q_z \cdot q_w \in A\) and \(q_z \cdot q_w \in B\) if \(z,w \in \R_+\). Further, for \(z,w \in \C_r\), we have

\begin{align*}
\left|q_z(\lambda) - q_w(\lambda)\right| &= \left|\frac{\lambda+1}{\lambda+z} - \frac{\lambda+1}{\lambda+w}\right| = \frac{|\lambda+1| \cdot |w-z|}{|\lambda+z| \cdot |\lambda+w|}
\\&= |q_z(\lambda)| \cdot \frac{|w-z|}{|\lambda+w|} \leq |q_z(\lambda)| \cdot \frac{|w-z|}{|w|} = |q_z(\lambda)| \cdot \left|1-\frac{z}{w}\right|,
\end{align*}

\[\left\lVert q_z - q_w\right\rVert_\infty \leq \left\lVert q_z\right\rVert_\infty \cdot \left|1-\frac{z}{w}\right| \xrightarrow{w \to z} 0.\]

This shows that, for \(z \in \C_r\), one has

\[q_z \cdot q_z = \lim_{\substack{w \to z \\ w \neq z}} q_z \cdot q_w \in A.\]

By the same argument we get \(q_z \cdot q_z \in B\) for all \(z \in \R_+\).
We therefore have

\[q_z \cdot q_w \in A \quad \forall z,w \in \C_r \qquad \text{and} \qquad q_z \cdot q_w \in B \quad \forall z,w \in \R_+\]

This implies that

\[A \cdot A \subeq A \quad \text{and} \quad B \cdot B \subeq B,\]

so that \(A\) and \(B\) are algebras. We have

\[\textbf{1} = q_1 \in B \subeq A.\]

Further, for \(\lambda,y \in [0,\infty]\) with \(\lambda \neq y\), we have

\[q_2(\lambda) = \frac{\lambda+1}{\lambda+2} \neq \frac{y+1}{y+2} = q_2(y),\]

so the algebras \(A\) and \(B\) separate points. Since

\[\overline{q_z} = q_{\overline{z}}, \quad \forall z \in \C_r\]

the algebra \(A\) is invariant under complex conjugation. Therefore, by the Stone–Weierstraß Theorem, we have

\[A = \overline{A} = C([0,\infty],\C) \quad \text{and} \quad B = \overline{B} = C([0,\infty],\R). \qedhere\]

Let \(\nu,\nu' \in \mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right)\). Then \(\Psi_\nu = \Psi_{\nu'}\), if and only if \(\nu = \nu'\).

Since the other implication is trivial, we assume that \(\Psi_\nu = \Psi_{\nu'}\) and show that then \(\nu = \nu'\). Setting

\[f_p: [0,\infty] \to \R, \quad \lambda \mapsto \frac{1+\lambda^2}{p^2+\lambda^2}, \qquad p \in \R^\times,\]

by assumption, we have

\[\int_{[0,\infty]} f_p \,d\nu = \Psi_\nu(p) = \Psi_{\nu'}(p) = \int_{[0,\infty]} f_p \,d\nu' \qquad \forall p \in \R^\times.\]

This yields that

\[\int_{[0,\infty]} f \,d\nu = \int_{[0,\infty]} f \,d\nu' \qquad \forall f \in \overline{\spann \{f_p: p \in \R^\times\}}.\]

Now, for \(z \in \R_+\), we consider the function

\begin{equation*}
q_z: [0,\infty] \to \R, \quad \lambda \mapsto \frac{\lambda+1}{\lambda+z}.
\end{equation*}

Then, defining the map

\[\cW: C([0,\infty],\R) \to C([0,\infty],\R), \quad f \mapsto f \circ \sqrt{\cdot},\]

for \(p \in \R^\times\), we get

\[(\cW f_p)(\lambda) = f_p\big(\sqrt{\lambda}\big) = \frac{1+\lambda}{p^2+\lambda} = q_{p^2}(\lambda), \quad \lambda \in \R_+,\]

so \(\cW f_p = q_{p^2}\). Using that \(\cW\) is an isometric isomorphism,
by prop:H2Dense0Infty, we get

\begin{align*}
C([0,\infty],\R) &= \cW^{-1} C([0,\infty],\R) = \cW^{-1} \overline{\spann_\R \left\lbrace q_z: z \in \R_+\right\rbrace}
\\&= \cW^{-1} \overline{\spann_\R \left\lbrace q_{p^2}: p \in \R^\times\right\rbrace} = \overline{\spann_\R \left\lbrace \cW^{-1} q_{p^2}: p \in \R^\times\right\rbrace} = \overline{\spann \{f_p: p \in \R^\times\}}.
\end{align*}

Therefore, we have

\[\int_{[0,\infty]} f \,d\nu = \int_{[0,\infty]} f \,d\nu' \quad \forall f \in C([0,\infty],\R),\]

which implies \(\nu = \nu'\).

Another ingredient we need to prove the injectivity of the map

\[\left(\mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\}\right) \slash \sim \,\ni [\nu] \mapsto h_\nu\]

is a better understanding of the kernel

\[\ker\left(\Theta_{h_\nu}-{\bf 1}\right) \cap H^2_\Lambda\left(\C_+\right)\]

appearing in prop:GeneralizedFixedPoint. This will be provided by the following proposition:

For every measure \(\nu \in \mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\}\), we have

\begin{equation*}
\ker\left(\Theta_{h_\nu}-{\bf 1}\right) \cap H^2_\Lambda\left(\C_+\right) = \C F_\nu.
\end{equation*}

For every \(C \in \C\) we have

\[\Theta_{h_\nu} (C \cdot F_\nu) = \frac{F_\nu}{RF_\nu} R(C \cdot F_\nu) = C \cdot \frac{F_\nu}{RF_\nu} RF_\nu = C \cdot F_\nu.\]

This yields

\[\C F_\nu \subeq \ker\left(\Theta_{h_\nu}-{\bf 1}\right) \cap H^2_\Lambda\left(\C_+\right).\]

We now show the other inclusion. For this we take a function \(F \in \ker\left(\Theta_{h_\nu}-{\bf 1}\right) \cap H^2_\Lambda\left(\C_+\right)\). We now want to show that \(F \in \C F_\nu\). If \(F=0\) the statement is trivial, so we can assume that \(F \neq 0\). By cor:kerNonTrivOuterFixExtended, we have

\[F \in \ker(\Theta_h - \textbf{1}) \cap \Out^2_\Lambda(\C_+)\]

and therefore, by prop:FixedPointFromConstruction there exists a measure \(\nu' \in \mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\}\) and \(C \in \T\) such that

\[F = C \cdot F_{\nu'}.\]

We notice that for the function

\[K(x) \coloneqq \frac 1{|F_\nu(x)| \cdot \sqrt{1+x^2}}, \quad x \in \R,\]

by lem:OuterExamples and lem:OuterHomo, we have

\[\Out(K)(z) = \frac 1{F_\nu(z)} \cdot \frac i{z+i}, \quad z \in \C_+.\]

Further, by lem:PsiEstimate, one has

\[K(x) = \frac 1{|F_\nu(x)| \cdot \sqrt{1+x^2}} = \frac 1{\sqrt{\Psi_\nu(x)} \cdot \sqrt{1+x^2}} \leq 1, \quad x \in \R,\]

so, by thm:OuterBetrag, we have \(\Out(K) \in H^\infty(\C_+)\). By lem:FNuLambdaOuter, we get

\[\Lambda F = C \cdot \Lambda F_{\nu'} \in H^2(\C_+)\]

and therefore, by prop:H2InclusionFunctions, we have

\[g \coloneqq \Out(K) \cdot \Lambda F \in H^2(\C_+).\]

\[g(z) = \frac 1{F_\nu(z)} \cdot \frac i{z+i} \cdot \frac{iz}{(z+i)^2} \cdot F(z) = -\frac z{(z+i)^3} \cdot \frac{F(z)}{F_\nu(z)}, \quad z \in \C_+.\]

Then, using that \(F,F_\nu \in \ker\left(\Theta_{h_\nu}-{\bf 1}\right)\), we get

\begin{align*}
\frac{F(x)}{F_\nu(x)} = \frac{h_\nu(x) \cdot (RF)(x)}{h_\nu(x) \cdot (RF_\nu)(x)} = \frac{F(-x)}{F_\nu(-x)}, \quad x \in \R
\end{align*}

and therefore

\begin{align*}
g(x) &= -\frac x{(x+i)^3} \cdot \frac{F(x)}{F_\nu(x)} = -\frac x{(x+i)^3} \cdot \frac{F(-x)}{F_\nu(-x)}
\\&= - \frac{(-x+i)^3}{(x+i)^3} \cdot g(-x) = \frac{(x-i)^3}{(x+i)^3} \cdot (Rg)(x) = {\phi_i(x)}^3 (Rg)(x), \quad x \in \R.
\end{align*}

Then, since \(Rg \in RH^2(\C_+) = H^2(\C_-)\), we get

\[g \in H^2(\C_+) \cap M_{\phi_i}^3 H^2(\C_-).\]

Now, we have

\begin{align*}
H^2(\C_+) \cap M_{\phi_i} H^2(\C_-) &= H^2(\C_+) \cap M_{\phi_i} \left(H^2(\C_+)^\perp\right) = \C Q_i,
\end{align*}

using lem:BlaschkeOrthogonal in the last step. This implies

\begin{align*}
H^2(\C_+) \cap M_{\phi_i}^3 H^2(\C_-) &= H^2(\C_+) \cap M_{\phi_i}^3 \left(H^2(\C_+)^\perp\right) = \bigoplus_{k=1}^3 M_{\phi_i}^{k-1}H^2(\C_+) \cap M_{\phi_i}^k \left(H^2(\C_+)^\perp\right)
\\&= \bigoplus_{k=1}^3 M_{\phi_i}^{k-1}\left(H^2(\C_+) \cap M_{\phi_i} \left(H^2(\C_+)^\perp\right)\right) = \bigoplus_{k=1}^3 M_{\phi_i}^{k-1}\C Q_i
\\&= \C Q_i + \C M_{\phi_i} Q_i + \C M_{\phi_i}^2 Q_i = (\C M_\textbf{1} + \C M_{\phi_i} + \C M_{\phi_i}^2) \,Q_i.
\end{align*}

Therefore, there are \(a,b,c \in \C\) such that

\[g(z) = (a + b \phi_i(z) + c \phi_i(z)^2) \cdot Q_i(z) = \left(a + b \left(\frac{z-i}{z+i}\right) + c \left(\frac{z-i}{z+i}\right)^2\right) \cdot \frac 1{2\pi} \cdot \frac i{z+i}, \quad z \in \C_+.\]

This is equivalent to

\[\frac{F(z)}{F_\nu(z)} = -\frac {(z+i)^3}z \cdot g(z) = \frac 1z \cdot \frac{-i}{2\pi} \left(a(z+i)^2 + b(z+i)(z-i) + c(z-i)^2\right), \quad z \in \C_+.\]

This shows that there are \(\alpha,\beta,\gamma \in \C\) such that

\[\frac{F(z)}{F_\nu(z)} = \frac 1z \cdot \left(\alpha z^2 + \beta z + \gamma\right) = \alpha z + \beta + \gamma\frac 1z, \quad z \in \C_+.\]

\[\alpha x + \beta + \gamma\frac 1x = \frac{F(x)}{F_\nu(x)} = \frac{F(-x)}{F_\nu(-x)} = -\alpha x + \beta -\gamma\frac 1x, \quad x \in \R^\times,\]

we have \(\alpha = \gamma = 0\) and therefore

\[\frac{F(z)}{F_\nu(z)} = \beta, \quad z \in \C_+,\]

so \(F = \beta \cdot F_\nu\).

The map

\[\left(\mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\}\right) \slash \sim \,\ni [\nu] \mapsto h_\nu\]

is injective.

Let \(\nu,\nu' \in \mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\}\) with \(h_\nu = h_{\nu'}\). Then, by prop:ker1Dim, we have

\[F_{\nu'} \in \ker\left(\Theta_{h_{\nu'}}-{\bf 1}\right) \cap H^2_\Lambda\left(\C_+\right) = \ker\left(\Theta_{h_\nu}-{\bf 1}\right) \cap H^2_\Lambda\left(\C_+\right) = \C F_\nu.\]

Since \(F_{\nu'} \neq 0\) this implies that there exists \(c \in \C^\times\) such that

\[F_{\nu'} = c \cdot F_\nu.\]

This yields

\[\Psi_{\nu'}(p) = |F_{\nu'}(p)|^2 = |c|^2 \cdot |F_\nu(p)|^2 = |c|^2 \cdot \Psi_\nu(p) = \Psi_{|c|^2 \cdot \nu}(p), \quad p \in \R.\]

This, by cor:PsiUnique, implies \(\nu' = |c|^2 \cdot \nu\) and therefore \(\nu' \sim \nu\), so \([\nu'] = [\nu]\).

§.§ The Osterwalder–Schrader transform of the Hardy space

In this section, given a function \(h \in L^\infty(\R,\T)^\flat\) such that the triple \(\left(L^2\left(\R,\C\right),H^2\left(\C_+\right),\theta_h\right)\) is a RPHS, we want to give an explicit form of the Osterwalder–Schrader transform (OS transform) of this RPHS, which is defined as follows:
(cf. <cit.>)
Let \(\left(\mathcal{E},\mathcal{E}_+,\theta\right)\) be a RPHS. We set

\begin{equation*}
\gls*{Ntheta} \coloneqq \left\lbrace \eta \in \mathcal{E}_+ : \braket*{\eta}{\theta\eta}=0\right\rbrace,
\end{equation*}

and consider the quotient map

\begin{equation*}
\gls*{qtheta}: \mathcal{E}_+ \to \mathcal{E}_+ \slash \,\mathcal{N}_\theta, \quad \eta \mapsto \eta + \cN_\theta.
\end{equation*}

On \(\mathcal{E}_+ \slash \,\mathcal{N}_\theta\) we define a scalar product by

\[\braket*{\eta + \cN_\theta}{\xi + \cN_\theta}_{\hat \cE} \coloneqq \braket*{\eta}{\theta \xi}, \quad \eta,\xi \in \cE_+\]

and write \(\gls*{Ehat}\) for the Hilbert space completion of \(\mathcal{E}_+ \slash \,\mathcal{N}_\theta\) with respect to the norm

\[\left\lVert \eta + \cN_\theta\right\rVert_{\hat{\cE}} \coloneqq \sqrt{\braket*{\eta + \cN_\theta}{\eta + \cN_\theta}_{\hat \cE}} = \sqrt{\braket*{\eta}{\theta\eta}}, \quad \eta \in \cE_+.\]

We call \(\hat{\mathcal{E}}\) the Osterwalder–Schrader transform of \(\cE_+\).

* Suppose that \(S : \mathcal{E}_+ \supseteq \mathcal{D}\left(S\right) \to \mathcal{E}_+\) is a linear
operator (not necessarily bounded) with

\[S\left(\mathcal{D}\left(S\right) \cap \mathcal{N}_\theta\right) \subseteq \mathcal{N}_\theta.\]

Then we define the linear operator

\begin{equation*}
\hat{S}: \mathcal{D}(\hat{S}) \coloneqq \hat{\mathcal{D}\left(S\right)} = \left\lbrace \hat{\xi}: \xi \in \mathcal{D}\left(S\right)\right\rbrace \to \hat{\mathcal{E}}, \quad \hat{S}\hat{\eta} \coloneqq \hat{S\eta}.
\end{equation*}

We call this operator \(\hat{S}\) the Osterwalder-Schrader transform of \(S\).

Let \(h \in L^\infty(\R,\T)^\flat\) such that the triple \(\left(L^2\left(\R,\C\right),H^2\left(\C_+\right),\theta_h\right) \eqqcolon (\cE,\cE_+,\theta)\) is a RPHS and let \(\mu \in \mathcal{BM}\left(\R_+\right)\) such that

\[\braket*{f}{\theta_h g} = \int_{\R_+} \overline{f(i\lambda)} g(i\lambda) \,d\mu(\lambda) \quad \forall f,g \in H^2(\C_+).\]

Further, for a function \(f \in H^2(\C_+)\), we consider the function \(f^\sV \in L^2\left(\R_+,\C,\mu\right)\) defined by

\[f^\sV(\lambda) = f(i\lambda), \quad \lambda \in \R_+.\]

Then the map

\begin{equation*}
\Phi: \cE_+ \big \slash \,\cN_\theta \to L^2\left(\R_+,\C,\mu\right), \quad f+\cN_\theta \mapsto f^\sV
\end{equation*}

extends to an isometric isomorphism of Hilbert spaces

\[\tilde \Phi: \hat \cE \to L^2\left(\R_+,\C,\mu\right).\]

For \(f \in H^2\left(\C_+\right)\) we have

\begin{equation*}
\left\lVert f^\sV\right\rVert^2 = \int_{\R_+} \overline{f(i\lambda)} f(i\lambda) \,d\mu(\lambda) = \braket*{f}{\theta_h f} = \braket*{f+\cN_\theta}{f+\cN_\theta}_{\hat \cE},
\end{equation*}

so the map \(\Phi\) is isometric and therefore well defined and injective. This also shows that \(\Phi\) maps every Cauchy sequence in \(\cE_+ \big \slash \,\cN_\theta\) to a Cauchy sequence in \(L^2\left(\R_+,\C,\mu\right)\) and therefore we can continue \(\Phi\) to a map \(\tilde \Phi: \hat \cE \to L^2\left(\R_+,\C,\mu\right)\) in a continuous way. It remains to show that this map \(\tilde \Phi\) is surjective. So let \(F \in \tilde \Phi(\hat \cE)^\perp\). Then, for every \(z \in \C_+\), we have

\begin{equation*}
0 = \braket{\Phi(Q_z + \cN_\theta)}{F} = \int_{\R_+} \overline{Q_z(i\lambda)} F(\lambda) \,d\mu(\lambda).
\end{equation*}

\[Q_z(i\lambda) = \frac 1{2\pi} \frac i{i\lambda-\overline{z}} = \frac 1{2\pi} \frac i{i\lambda+i} \cdot \frac{i\lambda + i}{i\lambda-\overline{z}} = Q_i(i\lambda) \cdot \frac{i\lambda + i}{i\lambda-\overline{z}} = Q_i(i\lambda) \cdot \frac{\lambda + 1}{\lambda+i\overline{z}}, \quad \lambda \in \R_+,\]

we have

\[0 = \int_{\R_+} \overline{Q_z(i\lambda)} F(\lambda) \,d\mu(\lambda) = \int_{\R_+} \frac{\lambda + 1}{\lambda-iz} \cdot \overline{Q_i(i\lambda)} F(\lambda) \,d\mu(\lambda) \quad \forall z \in \C_+.\]

Further, since \(z \in \C_+\) is equivalent to \((-iz) \in \C_r\), this is equivalent to

\[0 = \int_{\R_+} \frac{\lambda + 1}{\lambda+z} \cdot \overline{Q_i(i\lambda)} F(\lambda) \,d\mu(\lambda) \quad \forall z \in \C_r.\]

Then prop:H2Dense0Infty implies that

\[0 = \int_{\R_+} f(\lambda) \cdot \overline{Q_i(i\lambda)} F(\lambda) \,d\mu(\lambda) \quad \forall f \in C([0,\infty],\C),\]

so especially

\[0 = \int_{\R_+} f(\lambda) \cdot \overline{Q_i(i\lambda)} F(\lambda) \,d\mu(\lambda) \quad \forall f \in C_c(\R_+,\C).\]

Since the function \(\R_+ \ni \lambda \mapsto \overline{Q_i(i\lambda)}\) is continuous and bounded from below on compact intervals, this yields

\[0 = \int_{\R_+} f(\lambda) F(\lambda) \,d\mu(\lambda) \quad \forall f \in C_c(\R_+,\C),\]

Therefore, by <cit.>, we have \(F = 0 \in L^2\left(\R_+,\C,\mu\right)\). This shows that \(\tilde \Phi(\hat \cE)\) is dense in \(L^2\left(\R_+,\C,\mu\right)\), but since \(\tilde \Phi\) is isometric, we know that \(\tilde \Phi(\hat \cE)\) is a closed subspace of \(L^2\left(\R_+,\C,\mu\right)\) and therefore is equal to \(L^2\left(\R_+,\C,\mu\right)\).

We now want to apply this proposition to maximal RPHS. By thm:GeneralizedMaxPosForm we know that the functions \(h \in L^\infty(\R,\T)^\flat\) for which \(\left(L^2\left(\R,\C\right),H^2\left(\C_+\right),\theta_h\right)\) is a maximal RPHS, are precisely the functions of the form \(h = h_{\nu}\) for some measure \({\nu \in \mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\}}\). So we are interested in how the Osterwalder–Schrader transform looks in these cases:

Let \({\nu \in \mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\}}\) and \((\cE,\cE_+,\theta) \coloneqq \left(L^2\left(\R,\C\right),H^2\left(\C_+\right),\theta_{h_\nu}\right)\). Then

\[\hat \cE \cong L^2\left(\R_+,\C,\cT\nu\right),\]

where the isometric isomorphism between these Hilbert spaces is given by the map

\[\hat \cE \to L^2\left(\R_+,\C,\cT\nu\right), \quad f + \cN_\theta \mapsto f^\sV\]

\begin{equation*}
f^\sV(\lambda) \coloneqq f(i\lambda), \quad \lambda \in \R_+, f \in H^2(\C_+).
\end{equation*}

This follows immediately by thm:cTNuIsCarleson and prop:OsterwalderSchraderHardy.

§ APPLICATION TO HANKEL OPERATORS

In the last chapter, we answered the question for which functions \(h \in L^\infty(\R,\T)^\flat\) the triple \(\left(L^2\left(\R,\C\right),H^2\left(\C_+\right),\theta_h\right)\) is a maximal RPHS. The triple \(\left(L^2\left(\R,\C\right),H^2\left(\C_+\right),\theta_h\right)\) being a RPHS is, by lem:RPHSPosHankelEquiv, equivalent to the operator

\[H_h \coloneqq p_+ \theta_h p_+^* = p_+ M_h R \,p_+^*\]

being a positive operator, where by \(\gls*{p+}: L^2(\R,\C) \to H^2(\C_+)\) we denote the orthogonal projection from \(L^2(\R,\C)\) onto its subspace \(H^2(\C_+)\). We have already seen that this operator \(H_h\) is an example of a Hankel operator (cf. def:Hankel). These Hankel operators will be the object of investigation in this chapter. We start with the following definition:

* We write \(\gls*{Han}\) for the set of Hankel operators \(H \in B(H^2(\C_+))\) and write

\[\gls*{HanP} \coloneqq \{H \in \Han: H \geq 0\}\]

for the set of positive Hankel operators.

* For a function \({h \in L^\infty(\R,\C)}\) we define

\[\gls*{Hh} \coloneqq p_+ M_h R \,p_+^* \in B(H^2(\C_+)).\]

* A function \({h \in L^\infty(\R,\C)}\) is called a symbol of an Hankel operator \(H \in B(H^2(\C_+))\) if \(H = H_h\).

We have the following result about symbols of Hankel operators:
(Nehari's Theorem) (cf. <cit.>)
For every \({h \in L^\infty(\R,\C)}\) the operator \(H_h\) is a Hankel operator. Conversely, for every Hankel operator \(H\) there exists \({h \in L^\infty(\R,\C)}\) such that \(H = H_h\). Further, this symbol \(h\) can be choosen in a way such that \(\left\lVert h\right\rVert_\infty = \left\lVert H\right\rVert\).

Since the map

\[L^\infty(\R,\C) \ni h \mapsto H_h\]

is linear, the question whether a given Hankel operator has a unique symbol comes down to the question for which functions \(h \in L^\infty(\R,\C)\) one has \(H_h = 0\). This question is answered in the following proposition:
(cf. <cit.>)
For a function \(h \in L^\infty(\R,\C)\) one has \(H_h = 0\), if and only if \(h \in H^\infty\left(\C_-\right)\).

One has \(H_h = 0\), if and only if

\[\{0\} = p_+ M_h R p_+^* H^2(\C_+) = p_+ M_h H^2(\C_-),\]

which is equivalent to

\[M_h H^2(\C_-) \subeq H^2(\C_-).\]

This, by prop:H2InclusionFunctions, is equivalent to \(h \in H^\infty\left(\C_-\right)\).

We already know that, given a function \(h \in L^\infty(\R,\T)^\flat\), for which \(\left(L^2\left(\R,\C\right),H^2\left(\C_+\right),\theta_h\right)\) is a maximal RPHS, the Hankel operator \(H_h\) is positive and satisfies

\[\left\lVert H_h\right\rVert = \left\lVert p_+ M_h R p_+^*\right\rVert \leq \left\lVert M_h R \right\rVert = \left\lVert h \right\rVert_\infty = 1.\]

By thm:GeneralizedMaxPosForm we know that these functions are exactly the functions of the form \(h = h_{\nu}\) for some measure \(\nu \in \mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\}\). So we get a map

\[\mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\} \mapsto \{H \in \HanP : \left\lVert H\right\rVert \leq 1\}, \quad \nu \mapsto H_{h_\nu}.\]

The goal of this chapter is to show that this map is surjective or in other words: Not only does every maximal RPHS of the form \(\left(L^2\left(\R,\C\right),H^2\left(\C_+\right),\theta_h\right)\) give rise to a positive contractive Hankel operator, but in fact every positive contractive Hankel operator can be obtained in this manner. To prove this, we need to link Hankel operators to measures, which we will do in the following section.

§.§ Hankel operators, Carleson measures and the WOT

In this section we want to see that there is a one-to-one correspondence between positive Hankel operators and a certain class of measures, which are called Carleson measures and defined as follows:

* A measure \(\mu \in \mathcal{BM}^{\mathrm{fin}}\left(\R_+\right)\) is called Carleson measure, if the map

\[H^2(\C_+)^2 \to \C, \quad (f,g) \mapsto \int_{\R_+} \overline{f\left(i\lambda\right)}g\left(i\lambda\right) \,d\mu\left(\lambda\right)\]

defines a continuous sesquilinear form on \(H^2(\C_+)\).

* We write \(\CM\) for the set of Carleson measures on \(\R_+\).

* Given a Carleson measure \(\mu \in \CM\), we write \(\gls*{Hmu} \in B(H^2(\C_+))\) for the operator defined by

\[\braket*{f}{H_\mu g} \coloneqq \int_{\R_+} \overline{f\left(i\lambda\right)}g\left(i\lambda\right) \,d\mu\left(\lambda\right) \quad \forall f,g \in H^2(\C_+).\]

We start by showing that the so obtained operators \(H_\mu \in B(H^2(\C_+))\) with \(\mu \in \CM\) are Hankel operators:

Let \(\mu \in \CM\). Then \(H_\mu \in \HanP\).

For every \(t \in \R_+\) and \(f,g \in H^2(\Omega)\) we have

\begin{align*}
\braket*{f}{H_\mu S_t g} &= \int_{\R_+} \overline{f\left(i\lambda\right)}e^{it \cdot i\lambda} g\left(i\lambda\right) \,d\mu\left(\lambda\right) = \int_{\R_+} \overline{f\left(i\lambda\right)}e^{-t\lambda} g\left(i\lambda\right) \,d\mu\left(\lambda\right)
\\&= \int_{\R_+} \overline{e^{-t\lambda}f\left(i\lambda\right)} g\left(i\lambda\right) \,d\mu\left(\lambda\right) = \int_{\R_+} \overline{e^{it \cdot i\lambda}f\left(i\lambda\right)} g\left(i\lambda\right) \,d\mu\left(\lambda\right)
\\&= \braket*{S_t f}{H_\mu g} = \braket*{f}{S_t^* H_\mu g}
\end{align*}

so \(H S_t = S_t^* H\) and therefore \(H_\mu\) is a Hankel operator. That \(H_\mu\) is positive follows from the fact that, for \(f \in H^2(\Omega)\), we have

\[\braket*{f}{H_\mu f} = \int_{\R_+} \overline{f\left(i\lambda\right)}f\left(i\lambda\right) \,d\mu\left(\lambda\right) = \int_{\R_+} \left|f\left(i\lambda\right)\right|^2 \,d\mu\left(\lambda\right) \geq 0. \qedhere\]

The following result shows that in fact every positive Hankel operator can be obtained in this manner from a unique measure \(\mu \in \CM\):

The map

\[\CM \to \HanP, \quad \mu \mapsto H_\mu\]

is a bijection.

The surjectivity follows immediately by thm:HankelHasMeasure. We now show the injectivity. So let \(\mu,\mu' \in \CM\) with \(H_\mu = H_{\mu'}\). This is equivalent to

\[\int_{\R_+} \overline{f\left(i\lambda\right)}g\left(i\lambda\right) \,d\mu\left(\lambda\right) = \int_{\R_+} \overline{f\left(i\lambda\right)}g\left(i\lambda\right) \,d\mu'\left(\lambda\right) \quad \forall f,g \in H^2(\C_+),\]

which especially implies

\[\int_{\R_+} \overline{Q_i\left(i\lambda\right)}Q_z\left(i\lambda\right) \,d\mu\left(\lambda\right) = \int_{\R_+} \overline{Q_i\left(i\lambda\right)}Q_z\left(i\lambda\right) \,d\mu'\left(\lambda\right) \quad \forall z \in \C_+.\]

For \(z \in \C_+\), we have

\[Q_z\left(i\lambda\right) = \frac 1{2\pi} \cdot \frac i{i\lambda-\overline{z}} = \frac 1{2\pi} \cdot \frac 1{\lambda+i\overline{z}},\]

\[4\pi^2 \overline{Q_i\left(i\lambda\right)}Q_z\left(i\lambda\right) = \frac 1{\lambda +1} \cdot \frac 1{\lambda+i\overline{z}} = \frac {\lambda+1}{\lambda+i\overline{z}} \cdot \frac 1{(\lambda +1)^2}.\]

Therefore, since \(z \in \C_+\), if and only if \(i\overline{z} \in \C_r\), we have

\[\int_{\R_+} \frac {\lambda+1}{\lambda+z} \cdot \frac 1{(\lambda +1)^2} \,d\mu\left(\lambda\right) = \int_{\R_+} \frac {\lambda+1}{\lambda+z} \cdot \frac 1{(\lambda +1)^2} \,d\mu'\left(\lambda\right) \quad \forall z \in \C_r.\]

By prop:H2Dense0Infty this implies

\[\int_{\R_+} f(\lambda) \cdot \frac 1{(\lambda +1)^2} \,d\mu\left(\lambda\right) = \int_{\R_+} f(\lambda) \cdot \frac 1{(\lambda +1)^2} \,d\mu'\left(\lambda\right) \quad \forall f \in C([0,\infty],\C)\]

and therefore \(\mu = \mu'\).

This theorem shows that there is a one-to-one correspondence between positive Hankel operators and Carleson measures. Hence we can define:

Let \(H\) be a positive Hankel operator. Then we write \(\gls*{muH} \in \CM\) for the unique Carleson measure with \(H = H_{\mu_H}\).

Since we are interested in positive Hankel operators \(H\) of the form \(H = H_{h_\nu}\) with measures \({\nu \in \mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\}}\), we want to know what their corresponding Carleson measure is. The following proposition will answer this question:

Let \(\nu \in \mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\}\). Then \(H_{h_\nu} \in \HanP\) with \(\left\lVert H_{\nu_n}\right\rVert \leq 1\). Further \(\cT\nu \in \CM\) (cf. def:defCT) and

\[H_{\cT\nu} = H_{h_\nu}.\]

By thm:cTNuIsCarleson, for all \(f,g \in H^2(\C_+)\), one has

\[\int_{\R_+} \overline{f(i\lambda)}\,g(i\lambda) \,d(\cT \nu)(\lambda) = \braket*{f}{\theta_{h_\nu}g} = \braket*{f}{H_{h_\nu}g},\]

which immediately implies the statement.

Since later we will work with approximations in the weak operator topology, we now want to see how this topology behaves on the set of positive Hankel operators. For this we make the following definition:

Let \(\cH\) be a Hilbert space over a field \(\K = \R,\C\).

* Given a subset \(M \subeq B(\cH)\), we write

\[\overline{M}^{\gls*{WOT}}\]

for the closure of \(M\) with respect to the weak operator topology, i.e. the topology induced by the functions

\[f_{v,w}: B(\cH) \to \K, \quad A \mapsto \braket*{v}{Aw}, \quad v,w \in \cH.\]

* Given \(A \in B(\cH)\) and a sequence \((A_n)_{n \in \N}\) in \(B(\cH)\), we write

\[A_n \underset{\mathrm{WOT}}{\xrightarrow{n \to \infty}} A\]

if \(A_n\) converges to \(A\) with respect to the weak operator topology, i.e. if

\[\braket*{v}{A_n w} \xrightarrow{n \to \infty} \braket*{v}{Aw} \quad \forall v,w \in \cH.\]

Let \(\cH\) be a Hilbert space. Let \(A \in B(\cH)\) and \((A_n)_{n \in \N}\) be a sequence in \(B(\cH)\) with

\[A_n \underset{\mathrm{WOT}}{\xrightarrow{n \to \infty}} A.\]

\[\left\lVert A\right\rVert \leq \limsup_{n \to \infty} \left\lVert A_n \right\rVert.\]

For \(v,w \in \cH\) we have

\[|\braket*{v}{Aw}| = \limsup_{n \to \infty} |\braket*{v}{A_nw}| \leq \limsup_{n \to \infty} \left\lVert A_n \right\rVert \cdot \left\lVert v \right\rVert \cdot \left\lVert w \right\rVert,\]

which immediately implies the statement.

The set

\[\{H \in \HanP : \left\lVert H\right\rVert \leq 1\}\]

is closed in the weak operator topology.

Let \(H \in \overline{\{H \in \HanP : \left\lVert H\right\rVert \leq 1\}}\). Since \(H^2(\C_+)\) is separable, the weak operator topology restricted to the closed unit ball is metrizable and therefore there exists a sequence \((H_n)_{n \in \N}\) in \(\{H \in \HanP : \left\lVert H\right\rVert \leq 1\}\) with

\[H_n \underset{\mathrm{WOT}}{\xrightarrow{n \to \infty}} W.\]

By lem:NormSubContinuous we then get

\[\left\lVert H\right\rVert \leq \limsup_{n \to \infty} \left\lVert H_n \right\rVert \leq 1.\]

Further, for \(t \in \R_+\) and \(f,g \in H^2(\C_+)\), we have

\[\braket*{f}{HS_t g} = \lim_{n \to \infty} \braket*{f}{H_nS_t g} = \lim_{n \to \infty} \braket*{f}{S_t^*H_n g} = \lim_{n \to \infty} \braket*{S_t f}{H_n g} = \braket*{S_t f}{Hg} = \braket*{f}{S_t^* Hg},\]

so \(HS_t = S_t^* H\) and therefore \(H \in \Han\). Also, for every \(f \in H^2(\C_+)\), one has

\[\braket*{f}{Hf} = \lim_{n \to \infty} \braket*{f}{H_nf} \geq 0,\]

so \(H \in \HanP\).

We now want to see how convergence of positive Hankel operators in the weak operator topology translates to convergence of their corresponding Carleson measures. For this we need the following lemmata:

Let \(\cH\) be a Hilbert space and \(V \subeq \cH\) be a dense subspace. Then, for \(A \in B(\cH)\) and a bounded sequence \((A_n)_{n \in \N}\) in \(B(\cH)\), the following are equivalent:

* \(A_n \underset{\mathrm{WOT}}{\xrightarrow{n \to \infty}} A\).

* \(\braket*{v}{A_n w} \xrightarrow{n \to \infty} \braket*{v}{Aw} \quad \forall v,w \in V\).

That (a) implies (b) is trivial. We now assume that (b) holds and let \(v,w \in \cH\) and \(\epsilon > 0\). Further let

\[M \coloneqq \sup_{n \in \N} \left\lVert A_n-A\right\rVert < \infty.\]

Then there exist \(\tilde v, \tilde w \in V\) with

\[\left\lVert w-\tilde w\right\rVert < \frac{\epsilon}{3(M \left\lVert v\right\rVert + 1)} \quad \text{and} \quad \left\lVert v-\tilde v\right\rVert < \frac{\epsilon}{3(M \left\lVert \tilde w\right\rVert + 1)}.\]

Further, since

\[\braket*{\tilde v}{A_n \tilde w} \xrightarrow{n \to \infty} \braket*{\tilde v}{A\tilde w},\]

there exists \(N \in \N\) such that

\[\left|\braket*{\tilde v}{(A_n - A) \tilde w}\right| < \frac \epsilon 3 \quad \forall n \geq N.\]

Then, for \(n \geq N\), one has

\begin{align*}
\left|\braket*{v}{(A_n - A) w}\right| &= \left|\braket*{\tilde v}{(A_n - A) \tilde w} + \braket*{v-\tilde v}{(A_n - A) \tilde w}+ \braket*{v}{(A_n - A) (w-\tilde w)}\right|
\\&\leq \left|\braket*{\tilde v}{(A_n - A) \tilde w}\right| + \left|\braket*{v-\tilde v}{(A_n - A) \tilde w}\right| + \left|\braket*{v}{(A_n - A) (w-\tilde w)}\right|
\\&\leq \left|\braket*{\tilde v}{(A_n - A) \tilde w}\right| + \left\lVert v - \tilde v\right\rVert \cdot M \cdot \left\lVert \tilde w\right\rVert + \left\lVert v\right\rVert \cdot M \cdot \left\lVert w - \tilde w\right\rVert
\\&< \frac \epsilon 3 + \frac \epsilon 3 + \frac \epsilon 3 = \epsilon. \qedhere
\end{align*}

Let \(B\) be a Banach space and \(V \subeq B\) be a dense subspace. Then, for \(\phi \in B'\) and a bounded sequence \((\phi_n)_{n \in \N}\) in \(B'\), the following are equivalent:

* \(\phi_n \underset{w}{\xrightarrow{n \to \infty}} \phi\).

* \(\phi_n(v) \xrightarrow{n \to \infty} \phi(v) \quad \forall v \in V\).

That (a) implies (b) is trivial. We now assume that (b) holds and let \(v \in B\) and \(\epsilon > 0\). Further let

\[M \coloneqq \sup_{n \in \N} \left\lVert \phi_n\right\rVert < \infty.\]

Then there exists \(\tilde v \in V\) with

\[\left\lVert v-\tilde v\right\rVert < \frac{\epsilon}{2(M + 1)}.\]

Further, since

\[\phi_n(\tilde v) \xrightarrow{n \to \infty} \phi(\tilde v),\]

there exists \(N \in \N\) such that

\[\left|(\phi_n - \phi)(\tilde v)\right| < \frac \epsilon 2 \quad \forall n \geq N.\]

Then, for \(n \geq N\), one has

\begin{align*}
\left|(\phi_n - \phi)(v)\right| &= \left|(\phi_n - \phi)(\tilde v) + (\phi_n - \phi)(v-\tilde v)\right|
\\&\leq \left|(\phi_n - \phi)(\tilde v)\right| + \left|(\phi_n - \phi)(v-\tilde v)\right|
\\&\leq \left|(\phi_n - \phi)(\tilde v)\right| + M \cdot \left\lVert v - \tilde v\right\rVert < \frac \epsilon 2 + \frac \epsilon 2 = \epsilon. \qedhere
\end{align*}

For \(H \in \HanP\) and a bounded sequence \((H_n)_{n \in \N}\) in \(\HanP\) the following are equivalent:

* \(H_n \underset{\mathrm{WOT}}{\xrightarrow{n \to \infty}} H\).

* \(\braket*{Q_w}{H_n Q_z} \xrightarrow{n \to \infty} \braket*{Q_w}{H Q_z} \quad \forall w,z \in \C_+\).

* One has

\[\int_{\R_+} f(\lambda) \frac 1{(1+\lambda)^2}\,d\mu_{H_n}(\lambda) \xrightarrow{n \to \infty} \int_{\R_+} f(\lambda) \frac 1{(1+\lambda)^2}\,d\mu_H(\lambda)\]

for every function \(f \in C([0,\infty],\C)\).

* One has

\[\int_{\R_+} f(\lambda) \,d\mu_n(\lambda) \xrightarrow{n \to \infty} \int_{\R_+} f(\lambda) \,d\mu(\lambda)\]

for every function \(f \in C_c(\R_+,\C)\) and

\[\int_{\R_+} \frac 1{(1+\lambda)^2}\,d\mu_n(\lambda) \xrightarrow{n \to \infty} \int_{\R_+} \frac 1{(1+\lambda)^2}\,d\mu(\lambda).\]

(a) \(\Leftrightarrow\) (b): This follows immediately by the sesquilinearity of the scalar product, the fact that, by lem: evDense, the set \(\spann \{Q_z: z \in \C_+\}\) is dense in \(H^2(\C_+)\) and lem:WOTDenseFunctionals.

(b) \(\Leftrightarrow\) (c):
For \(z \in \C_+\), we have

\[Q_z\left(i\lambda\right) = \frac 1{2\pi} \cdot \frac i{i\lambda-\overline{z}} = \frac 1{2\pi} \cdot \frac 1{\lambda+i\overline{z}}, \quad \lambda \in \R_+.\]

For \(w,z \in \C_+\) this yields

\[\overline{Q_w\left(i\lambda\right)}Q_z\left(i\lambda\right) = \frac 1{4\pi^2 } \cdot \frac 1{\lambda-iw} \cdot \frac 1{\lambda+i\overline{z}} = \frac 1{4\pi^2 } \cdot \frac {\lambda + 1}{\lambda-iw} \cdot \frac {\lambda+1}{\lambda+i\overline{z}} \cdot \frac 1{(\lambda +1)^2}, \quad \lambda \in \R_+,\]

so setting

\[f_{w,z}: [0,\infty] \to \C, \quad \lambda \mapsto \frac 1{4\pi^2 } \cdot \frac {\lambda + 1}{\lambda-iw} \cdot \frac {\lambda+1}{\lambda+i\overline{z}},\]

we have \(f_{w,z} \in C([0,\infty],\C)\) and

\[\overline{Q_w\left(i\lambda\right)}Q_z\left(i\lambda\right) = f_{w,z}(\lambda) \cdot \frac 1{(\lambda +1)^2}, \quad \lambda \in \R_+.\]

\[f_{i,z}(\lambda) = \frac 1{4\pi^2 } \cdot \frac {\lambda + 1}{\lambda-i \cdot i} \cdot \frac {\lambda+1}{\lambda+i\overline{z}} = \frac 1{4\pi^2 } \cdot \frac {\lambda+1}{\lambda+i\overline{z}},\]

by prop:H2Dense0Infty, we have

\[\overline{\spann \{f_{w,z}: w,z \in \C_+\}} = C([0,\infty],\C),\]

where the closure is taken with respect to the supremum norm. The equivalence then follows by lem:WeakDenseFunctionals, since we have

\[\overline{Q_i\left(i\lambda\right)}Q_i\left(i\lambda\right) = f_{i,i}(\lambda) \cdot \frac 1{(\lambda +1)^2} = \frac 1{4\pi^2 } \cdot \frac 1{(\lambda +1)^2}\]

and therefore

\begin{align*}
\sup_{n \in \N} \,\int_{\R_+} \frac 1{(1+\lambda)^2}\,d\mu_n(\lambda) &= 4\pi^2 \cdot \sup_{n \in \N}\, \int_{\R_+} \overline{Q_i\left(i\lambda\right)}Q_i\left(i\lambda\right)\,d\mu_n(\lambda)
\\&= 4\pi^2 \cdot \sup_{n \in \N}\,\braket*{Q_i}{H_n Q_i} \leq 4\pi^2 \cdot \sup_{n \in \N} \left\lVert H_n\right\rVert \cdot \left\lVert Q_i\right\rVert^2 < \infty.
\end{align*}

(c) \(\Leftrightarrow\) (d): By the Stone–Weierstraß Theorem we have

\[\overline{C_c(\R_+,\C) \oplus (\C \cdot \textbf{1})} = C([0,\infty],\C),\]

where the closure is taken with respect to the supremum norm. This, by lem:WeakDenseFunctionals, implies that (c) holds, if and only if

\[\int_{\R_+} \frac 1{(1+\lambda)^2}\,d\mu_n(\lambda) \xrightarrow{n \to \infty} \int_{\R_+} \frac 1{(1+\lambda)^2}\,d\mu(\lambda)\]

\[\int_{\R_+} f(\lambda) \frac 1{(1+\lambda)^2} \,d\mu_n(\lambda) \xrightarrow{n \to \infty} \int_{\R_+} f(\lambda) \frac 1{(1+\lambda)^2} \,d\mu(\lambda) \quad \forall f \in C_c(\R_+,\C).\]

The latter is equivalent to

\[\int_{\R_+} f(\lambda) \,d\mu_n(\lambda) \xrightarrow{n \to \infty} \int_{\R_+} f(\lambda) \,d\mu(\lambda) \quad \forall f \in C_c(\R_+,\C),\]

since the function

\[\R_+ \ni \lambda \mapsto \frac 1{(1+\lambda)^2}\]

is continuous and bounded from below on compact sets.

As mentioned before, our goal is to show that the map

\[\mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\} \mapsto \{H \in \HanP : \left\lVert H\right\rVert \leq 1\}, \quad \nu \mapsto H_{h_\nu}\]

is surjective. We will do this in two steps. In the next section we will show that

\[\{H \in \HanP : \left\lVert H\right\rVert \leq 1\} = \overline{\left\{H_{h_{W(\mu)}}: \mu \in \mathcal{BM}^{\mathrm{fin}}\left(\R_+\right) \setminus \{0\}\right\}}^{\mathrm{WOT}}\]

and in the following section we then will use this to show that

\[\{H \in \HanP : \left\lVert H\right\rVert \leq 1\} = \left\{H_{h_{\nu}}: \nu \in \mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\}\right\}.\]

§.§ Hankel operators with fixed points and a density theorem

In this section we start by considering positive Hankel operators \(H \in \HanP\) with \(\left\lVert H\right\rVert \leq 1\) and with the additional assumption that \(H\) has a fixed point. A characterization of these operators is given by the following proposition:

Let \(H \in \HanP\) with \(\left\lVert H\right\rVert \leq 1\). Then the following are equivalent:

* \(\ker(H-\textbf{1}) \neq \{0\}\).

* There exists a function \(h \in L^\infty(\R,\T)^\flat\) such that \(H_h = H\) and \(\ker\left(\theta_h-{\bf 1}\right) \cap H^2(\C_+) \neq \left\lbrace 0\right\rbrace\).

* There exists a measure \(\mu \in \mathcal{BM}^{\mathrm{fin}}\left(\R_+\right) \setminus \{0\}\) such that \(H = H_{h_{W(\mu)}}\).

(a) \(\Rightarrow\) (b): Let \(f \in \ker(H-\textbf{1}) \setminus \{0\}\). By Nehari's theorem (thm:Nehari) there exists a function \(h \in L^\infty(\R,\C)\) with \(\left\lVert h \right\rVert_\infty = \left\lVert H\right\rVert \leq 1\) such that \(H_h = H\). Then, denoting by \(P_+\) the orthogonal projection onto \(H^2(\C_+)\), we have

\[\left\lVert f\right\rVert^2 = \left\lVert H_hf\right\rVert^2 = \left\lVert P_+hRf\right\rVert^2 = \left\lVert hRf\right\rVert^2 - \left\lVert (\textbf{1}-P_+)hRf\right\rVert^2 \leq \left\lVert f\right\rVert^2 - \left\lVert (\textbf{1}-P_+)hRf\right\rVert^2,\]

which yields \(\left\lVert (\textbf{1}-P_+)hRf\right\rVert^2 = 0\) and therefore \((\textbf{1}-P_+)hRf = 0\), so we get

\[\theta_h f = hRf = P_+hRf = H_hf = f.\]

This yields \(f \in \ker\left(\theta_h-{\bf 1}\right) \cap H^2(\C_+)\) and therefore

\[\ker\left(\theta_h-{\bf 1}\right) \cap H^2(\C_+) \neq \left\lbrace 0\right\rbrace.\]

Further, using thm:HardyAlmostEverywhereNonZero, we get

\[h = \frac{f}{Rf}.\]

This yields

\[h = \frac{f}{Rf} = R\left(\frac{Rf}{f}\right) = R\left(\frac{1}{h}\right) = \frac 1{Rh}.\]

Then, for almost every \(x \in \R\), we get

\[1 \geq \left\lVert h\right\rVert_\infty \geq \left|h(x)\right| = \frac{1}{|h(-x)|} \geq \frac 1{\left\lVert h\right\rVert_\infty} \geq 1\]

and therefore \(h(x) \in \T\), so we have \(h \in L^\infty(\R,\T)\). This yields \(\frac 1h = \overline{h}\) and therefore

\[h^\flat = \overline{Rh} = \overline{\left(\frac 1h\right)} = h.\]

(b) \(\Rightarrow\) (a): Let \(h \in L^\infty(\R,\T)^\flat\) and \(f \in \left(\ker\left(\theta_h-{\bf 1}\right) \cap H^2(\C_+)\right) \setminus \{0\}\). Then

\begin{equation*}
H f = H_h f = p_+ \theta_h p_+^* f = p_+ \theta_h f = p_+ f = f
\end{equation*}

and therefore \(f \in \ker(H-\textbf{1}) \setminus \{0\}\).

(b) \(\Leftrightarrow\) (c): By lem:RPHSPosHankelEquiv, for a function \(h \in L^\infty(\R,\T)^\flat\), one has \(H_h \in \HanP\), if and only if \((L^2(\R,\C),H^2(\C_+),\theta_h)\) is a RPHS. Further, by thm:thetaFixedPointCharac, the functions \(h \in L^\infty(\R,\T)^\flat\) for which \((L^2(\R,\C),H^2(\C_+),\theta_h)\) is a RPHS and \(\ker\left(\theta_h-{\bf 1}\right) \cap H^2(\C_+) \neq \left\lbrace 0\right\rbrace\) are precisely the functions of the form \(h_{W(\mu)}\) for some measure \(\mu \in \mathcal{BM}^{\mathrm{fin}}\left(\R_+\right) \setminus \{0\}\).

We now want to find a sufficient condition for a Hankel operator to have a fixed point, that is easy to check. One of these conditions is that the Hankel operator is compact:

Let \(H\) be a compact positive Hankel operator with \(\left\lVert H \right\rVert = 1\). Then

\[\ker(H-\textbf{1}) \neq \{0\}.\]

By the spectral theorem for compact self-adjoint operators there is a null sequence \(\left(\lambda_k\right)_{k \in \N}\) in \(\R_{\geq 0}\) with

\[\lambda_{k+1} \leq \lambda_k \quad \forall k \in \N\]

and a sequence of orthogonal finite dimensional projections \(\left(P_k\right)_{k \in \N}\) in \(B(H^2(\C_+))\) such that

\[H = \sum_{k = 1}^\infty \lambda_k P_k.\]

Since \(\left\lVert H \right\rVert = 1\), we have \(\lambda_1 = 1\) and \(P_1 \neq 0\). Therefore we can choose a function \(f \in P_1 H^2(\C_+) \setminus \{0\}\) and get

\[Hf = \lambda_1P_1 f = f. \qedhere\]

A powerful tool in checking if a Hankel operator is compact is given by Hartman's Theorem:
(Hartman's Theorem for the upper half plane)
A Hankel operator \(H\) is compact, if and only if \(H\) has a continuous symbol \(h \in C(\R)\) with

\[\lim_{p \to -\infty}h(p) = \lim_{p \to \infty}h(p).\]

This follows immediately by transforming Hartman's Theorem (<cit.>) from the unit disc to the upper half plane.

In general it is quite difficult to find a concrete symbol for a given Hankel operator, even moreso to find a continuous one as it is required in Hartman's Theorem, but for positive Hankel operators the following theorem provides concrete symbols with good analytic properties:
(cf. <cit.>)
Let \(H\) be a positive Hankel operator and set

\[h\left(p\right) \coloneqq \frac i\pi \cdot \int_{\R_+} \frac {p}{\lambda^2+p^2} \,d\mu_H\left(\lambda\right), \quad p \in \R.\]

Then \(h \in L^\infty\left(\R,\C\right)\) and \(H_h = H\).

This result together with Hartman's Theorem allows us to provide an easy to check criterion for a positive Hankel operator to be compact:

Let \(\mu \in \mathcal{BM}\left(\R_+\right)\) with

\[\int_{\R_+} \frac 1\lambda \,d\mu(\lambda)<\infty.\]

Then \(\mu \in \CM\) and the Hankel operator \(H_\mu\) is compact.

For \(f \in H^2(\C_+)\) and \(\lambda \in \R_+\), by prop:H2RKHS, we have

\[|f(i\lambda)| = |\braket*{Q_{i\lambda}}{f}| \leq \left\lVert Q_{i\lambda}\right\rVert \cdot \left\lVert f\right\rVert = \frac 1{2\sqrt{\pi \lambda}} \cdot \left\lVert f\right\rVert.\]

Therefore, for \(f,g \in H^2(\C_+)\), we get

\[\int_{\R_+} |f(i\lambda)| \cdot |g(i\lambda)| \,d\mu(\lambda) \leq \int_{\R_+} \frac 1{2\sqrt{\pi \lambda}} \cdot \left\lVert f\right\rVert \cdot \frac 1{2\sqrt{\pi \lambda}} \cdot \left\lVert g\right\rVert \,d\mu(\lambda) = \frac 1{4\pi} \int_{\R_+} \frac 1\lambda \,d\mu(\lambda) \cdot \left\lVert f\right\rVert \cdot \left\lVert g\right\rVert,\]

which shows that \(\mu \in \CM\). Further, by thm:CarlesonRepresentant, we know that, for the function

\[h(p) \coloneqq \frac i\pi \cdot \int_{\R_+} \frac{p}{\lambda^2+p^2} \,d\mu(\lambda),\]

we have \(h \in L^\infty(\R,\C)\) and \(H_h = H_\mu\). We now show that \(h \in C(\R)\) with

\[\lim_{p \to -\infty}h(p) = \lim_{p \to \infty}h(p),\]

since then the statement follows by thm:Hartman. For every \(\lambda \in \R_+\) and \(p \in \R\), we have

\[\left|\frac{p}{\lambda^2+p^2}\right| = \frac{\left|p\right|}{\lambda^2+\left|p\right|^2} \leq \frac{\left|p\right|}{2\lambda \left|p\right|} = \frac 1{2\lambda}.\]

Then, by the Dominated Convergence Theorem, for every \(p \in \R\), one has

\[\lim_{p' \to p} h(p') = \lim_{p' \to p} \frac i\pi \cdot \int_{\R_+} \frac{p'}{\lambda^2+p'^2} \,d\mu(\lambda) = \frac i\pi \cdot \int_{\R_+} \frac{p}{\lambda^2+p^2} \,d\mu(\lambda) = h(p),\]

so \(h \in C(\R)\). Also, by the Dominated Convergence Theorem, we get

\[\lim_{p \to \pm\infty} h(p) = \lim_{p \to \pm\infty} \frac i\pi \cdot \int_{\R_+} \frac{p}{\lambda^2+p^2} \,d\mu(\lambda) = \frac i\pi \cdot \int_{\R_+} 0 \,d\mu(\lambda) = 0,\]

which implies the statement.

Let \(H\) be a positive Hankel operator with \(\left\lVert H\right\rVert = 1\). Then there exists a sequence \((\mu_n)_{n \in \N}\) in \(\mathcal{BM}^{\mathrm{fin}}\left(\R_+\right) \setminus \{0\}\) such that

\[H_{h_{W(\mu_n)}} \underset{\mathrm{WOT}}{\xrightarrow{n \to \infty}} H.\]

For \(t \in \left[1,\infty\right)\) we define the measure \(\rho_t\) on \(\R_+\) by

\begin{equation*}
\rho_t(A) \coloneqq \mu_H\left(A \cap \left[\frac 1 t,t\right]\right)
\end{equation*}

for every Borel set \(A \subeq \R_+\). Then

\[\int_{\R_+} \frac 1\lambda \,d\rho_t(\lambda) = \int_{\left[\frac 1t,t\right]} \frac 1\lambda \,d\mu_H(\lambda) \leq \int_{\left[\frac 1t,t\right]} t \,d\mu_H(\lambda) = t \cdot \mu_H\left(\left[\frac 1t,t\right]\right) < \infty,\]

so, by prop:compactCriterion, we have \(\rho_t \in \CM\) and the Hankel operator \(H_{\rho_t}\) is compact. Further, for \(f,g \in H^2(\C_+)\) and \(t \in [1,\infty)\), one has

\begin{align*}
\left|\braket*{f}{(H-H_{\rho_t})g}\right| &= \left|\int_{\R \setminus \left[\frac 1 t,t\right]} \overline{f(i\lambda)}g(i\lambda)\,d\mu_H(\lambda)\right|
\\&\leq \sqrt{\int_{\R \setminus \left[\frac 1 t,t\right]} \left|f(i\lambda)\right|^2\,d\mu_H(\lambda)} \cdot \sqrt{\int_{\R \setminus \left[\frac 1 t,t\right]} \left|g(i\lambda)\right|^2\,d\mu_H(\lambda)} \xrightarrow{t \to \infty} 0.
\end{align*}

This implies that \(H_{\rho_t} \underset{\mathrm{WOT}}{\xrightarrow{t \to \infty}} H\). We have

\[\braket*{f}{H_{\rho_t}f} = \int_{\left[\frac 1 t,t\right]} \left|f(i\lambda)\right|^2\,d\mu_H(\lambda) \leq \int_{\R_+} \left|f(i\lambda)\right|^2\,d\mu_H(\lambda) = \braket*{f}{Hf},\]

and therefore \(\left\lVert H_{\rho_t} \right\rVert \leq \left\lVert H \right\rVert\). This, by lem:NormSubContinuous, yields

\[1 = \left\lVert H\right\rVert \leq \limsup_{t \to \infty}\left\lVert H_{\rho_t}\right\rVert \leq \limsup_{t \to \infty}\left\lVert H\right\rVert = 1.\]

Therefore, we can choose a sequence \((t_n)_{n \in \N}\) such that

\[t_n \xrightarrow{n \to \infty} 0 \quad \text{and} \quad \left\lVert H_{\rho_{t_n}}\right\rVert \xrightarrow{n \to \infty} 1\]

\[H_{\rho_{t_n}} \neq 0 \quad \forall n \in \N.\]

We now set

\[H_n \coloneqq \frac 1{\left\lVert H_{\rho_{t_n}}\right\rVert}H_{\rho_{t_n}}.\]

\[H_n \underset{\mathrm{WOT}}{\xrightarrow{n \to \infty}} H.\]

Further, for every \(n \in \N\), the operator \(H_n\) is a positive compact Hankel operator with \(\left\lVert H_n\right\rVert = 1\), so, by lem:compactFixpoint and prop:HankelFixedPoint, there exists a measure \(\mu_n \in \mathcal{BM}^{\mathrm{fin}}\left(\R_+\right) \setminus \{0\}\) such that \(H_n = H_{h_{W(\mu_n)}}\). We then have

\[H_{h_{W(\mu_n)}} = H_n \underset{\mathrm{WOT}}{\xrightarrow{n \to \infty}} H. \qedhere\]

One has

\[\{H \in \HanP : \left\lVert H\right\rVert \leq 1\} = \overline{\{H_{h_{W(\mu)}}: \mu \in \mathcal{BM}^{\mathrm{fin}}\left(\R_+\right) \setminus \{0\}\}}^{\mathrm{WOT}}.\]

By prop:cTNuIsCarlesonOfHNu we have

\[\{H_{h_{W(\mu)}}: \mu \in \mathcal{BM}^{\mathrm{fin}}\left(\R_+\right) \setminus \{0\}\} \subeq \{H \in \HanP : \left\lVert H\right\rVert \leq 1\}\]

and therefore, using lem:ContractiveHankelWOTClosed, we get

\begin{align*}
\overline{\{H_{h_{W(\mu)}}: \mu \in \mathcal{BM}^{\mathrm{fin}}\left(\R_+\right) \setminus \{0\}\}}^{\mathrm{WOT}} &\subeq \overline{\{H \in \HanP : \left\lVert H\right\rVert \leq 1\}}^{\mathrm{WOT}}
\\&= \{H \in \HanP : \left\lVert H\right\rVert \leq 1\}.
\end{align*}

We now show the other inclusion. For every \(x \in \R_+\) and every \(f,g \in H^2(\C_+)\), we have

\[\braket*{f}{H_{\delta_x} g} = \int_{\R_+}\overline{f(i\lambda)}g(i\lambda)\,d\delta_x(\lambda) = \overline{f(ix)}g(ix) = \braket{f}{Q_{ix}}\braket*{Q_{ix}}{g}.\]

Then, by prop:H2RKHS, we have

\[\left\lVert H_{\delta_x}\right\rVert = \left\lVert Q_{ix}\right\rVert^2 = \frac 1{4\pi x}.\]

Therefore, for the measure

\[\rho_x \coloneqq 4\pi x \cdot \delta_x,\]

we have \(\left\lVert H_{\rho_x}\right\rVert = 1\). Now, let \(H \in \HanP\) with \(\left\lVert H\right\rVert \leq 1\). Then, for every \(f \in H^2(\C_+)\), we have

\[\braket*{f}{(H+H_{\rho_x})f} = \braket*{f}{H f} + \braket*{f}{H_{\rho_x}f} \geq \braket*{f}{H_{\rho_x} f},\]

\[\left\lVert H+H_{\rho_x}\right\rVert \geq \left\lVert H_{\rho_x}\right\rVert = 1.\]

Therefore, for the function

\[F_x: [0,1] \to \R, \quad t \mapsto \left\lVert H+t \cdot H_{\rho_x}\right\rVert,\]

we have

\[F_x(0) = \left\lVert H\right\rVert \leq 1 \quad \text{and} \quad F_x(1) = \left\lVert H+H_{\rho_x}\right\rVert \geq 1,\]

so, by the Intermediate Value Theorem, there exists \(t_x \in [0,1]\) such that

\[1 = F_x(t_x) = \left\lVert H+t_x \cdot H_{\rho_x}\right\rVert.\]

This, by cor:strongDense, implies

\[H+t_x \cdot H_{\rho_x} \in \overline{\{H_{h_{W(\mu)}}: \mu \in \mathcal{BM}^{\mathrm{fin}}\left(\R_+\right) \setminus \{0\}\}}^{\mathrm{WOT}}, \quad \forall x \in \R_+.\]

Further, for \(w,z \in \C_+\), we have

\begin{align*}
\braket*{Q_w}{H_{\rho_x}Q_z} &= 4\pi x \cdot \int_{\R_+}\overline{Q_w(i\lambda)}Q_z(i\lambda)\,d\delta_x(\lambda)
\\&= 4\pi x \cdot \overline{Q_w(ix)}Q_z(ix) = \frac x{\pi} \cdot \overline{\frac 1{ix - \overline{w}}}\frac 1{ix - \overline{z}} \xrightarrow{x \to 0} 0,
\end{align*}

so, by prop:topologiesEquiv, we have

\[H_{\rho_x} \underset{\mathrm{WOT}}{\xrightarrow{x \to 0}} 0\]

and therefore

\[H + t_x \cdot H_{\rho_x} \underset{\mathrm{WOT}}{\xrightarrow{x \to 0}} H,\]

which shows that

\[H \in \overline{\{H_{h_{W(\mu)}}: \mu \in \mathcal{BM}^{\mathrm{fin}}\left(\R_+\right) \setminus \{0\}\}}^{\mathrm{WOT}}.\]

We therefore have

\[\{H \in \HanP : \left\lVert H\right\rVert \leq 1\} \subeq \overline{\{H_{h_{W(\mu)}}: \mu \in \mathcal{BM}^{\mathrm{fin}}\left(\R_+\right) \setminus \{0\}\}}^{\mathrm{WOT}}. \qedhere\]

§.§ Convergence in the weak-\(*\)-topology

In this section we want to better understand the set

\[\left\{H_{h_{\nu}}: \nu \in \mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\}\right\}.\]

For this we have to know, given a measure \(\nu \in \mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\}\) and a sequence \((\nu_n)_{n \in \N}\) in \(\mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\}\) with

\[\nu_n \underset{w}{\xrightarrow{n \to \infty}} \nu,\]

what are the consequences for the measures \(\cT \nu\) and \((\cT \nu_n)_{n \in \N}\). We start by the following definition:

For a measure \(\nu \in \mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\}\) we define the continuous function

\[\gls*{tnu}: \R_+ \to \R_+, \quad t_\nu(\lambda) = \left|F_\nu(i\lambda)\right|^{-2} \frac{1+\lambda^2}{\lambda}\]

with \(F_\nu\) as in def:FNuHNu.

By definition, for \(\nu \in \mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\}\), we have

\[d(\cT\nu)(\lambda) = t_\nu(\lambda)\,d\nu(\lambda)\]

(cf. def:defCT).

Let \(\nu \in \mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\}\) and \(\left(\nu_n\right)_{n \in \N}\) be a sequence in \(\mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\}\) with

\[\nu_n \underset{w}{\xrightarrow{n \to \infty}} \nu.\]

Then the functions \((t_{\nu_n})_{n \in \N}\) converge to \(t_\nu\) uniformly on compact sets.

By lem:OuterHomo, we have

\[F_\nu^2 = \Out\left(\sqrt{\Psi_\nu}\right)^2 = \Out(\Psi_\nu)\]

and analogously \(F_{\nu_n}^2 = \Out(\Psi_{\nu_n})\) for \(n \in \N\), so, by lem:OuterSymmetric, for every \(\lambda \in \R_+\), one has

\begin{equation*}
t_\nu(\lambda) = \frac{1+\lambda^2}{\lambda} \exp\left(-\frac{1}{\pi} \int_\R \frac{\lambda}{p^2+\lambda^2} \log\left(\Psi_\nu\left(p\right)\right) dp\right)
\end{equation*}

\begin{equation*}
t_{\nu_n}(\lambda) = \frac{1+\lambda^2}{\lambda} \exp\left(-\frac{1}{\pi} \int_\R \frac{\lambda}{p^2+\lambda^2} \log\left(\Psi_{\nu_n}\left(p\right)\right) dp\right).
\end{equation*}

Therefore, it suffices to show that

\[\int_\R \frac{\lambda}{p^2+\lambda^2} \log\left(\Psi_{\nu_n}\left(p\right)\right) dp \xrightarrow{n \to \infty} \int_\R \frac{\lambda}{p^2+\lambda^2} \log\left(\Psi_\nu\left(p\right)\right) dp\]

uniformly on \([a,b]\) with \(0<a<b<\infty\). Since, for every \(\lambda \in [a,b]\), one has

\begin{align*}
&\left|\int_\R \frac{\lambda}{p^2+\lambda^2} \log\left(\Psi_\nu\left(p\right)\right) dp - \int_\R \frac{\lambda}{p^2+\lambda^2} \log\left(\Psi_{\nu_n}\left(p\right)\right) dp\right|
\\&\qquad\qquad\qquad\qquad\leq \int_\R \frac{\lambda}{p^2+\lambda^2} \left|\log\left(\Psi_\nu\left(p\right)\right) - \log\left(\Psi_{\nu_n}\left(p\right)\right)\right| dp
\\&\qquad\qquad\qquad\qquad\leq \int_\R \frac{b}{p^2+a^2} \left|\log\left(\Psi_\nu\left(p\right)\right) - \log\left(\Psi_{\nu_n}\left(p\right)\right)\right| dp
\\&\qquad\qquad\qquad\qquad\leq b \cdot \max\left\{1,\frac 1{a^2}\right\} \cdot \int_\R \frac{1}{1+p^2} \left|\log\left(\Psi_\nu\left(p\right)\right) - \log\left(\Psi_{\nu_n}\left(p\right)\right)\right| dp,
\end{align*}

it remains to show that

\begin{equation*}
\int_\R \frac{1}{1+p^2} \left|\log\left(\Psi_\nu\left(p\right)\right) - \log\left(\Psi_{\nu_n}\left(p\right)\right)\right| dp \xrightarrow{n \to \infty} 0.
\end{equation*}

Since by assumption

\[\nu_n([0,\infty]) = \int_{[0,\infty]} \textbf{1} \,d\nu_n \xrightarrow{n \to \infty} \int_{[0,\infty]} \textbf{1} \,d\nu = \nu([0,\infty]) \neq 0,\]

we have that

\[M \coloneqq \sup_{n \in \N} \,\left|\log\left(\nu_n([0,\infty])\right)\right| < \infty.\]

Then, for \(n \in \N\) and \(p \in \R^\times\), by lem:PsiEstimate, we have

\[\left|\log\left(\Psi_{\nu_n}\left(p\right)\right)\right| = \left|\log\left(\nu_n([0,\infty])\right)\right| + \left|\log\left(\frac{\Psi_{\nu_n}\left(p\right)}{\nu_n([0,\infty])}\right)\right| \leq M + \left|\log\left(p^2\right)\right|\]

and analogously

\[\left|\log\left(\Psi_\nu\left(p\right)\right)\right| \leq M + \left|\log\left(p^2\right)\right|.\]

For \(n \in \N\) and \(p \in \R^\times\) this yields

\[\frac{1}{1+p^2} \left|\log\left(\Psi_\nu\left(p\right)\right) - \log\left(\Psi_{\nu_n}\left(p\right)\right)\right| \leq \frac{\left|\log\left(\Psi_\nu\left(p\right)\right)\right| + \left|\log\left(\Psi_{\nu_n}\left(p\right)\right)\right|}{1+p^2} \leq 2 \cdot \frac{M + \left|\log\left(p^2\right)\right|}{1+p^2}\]

Then, since

\[\int_\R 2 \cdot \frac{M + \left|\log\left(p^2\right)\right|}{1+p^2} \,dp < \infty,\]

and since, by lem:PsiPointwise, we have

\[\left|\log\left(\Psi_\nu\left(p\right)\right) - \log\left(\Psi_{\nu_n}\left(p\right)\right)\right| \xrightarrow{n \to \infty} 0, \quad \forall p \in \R^\times,\]

by the Dominated Convergence Theorem, we get

\begin{equation*}
\int_\R \frac{1}{1+p^2} \left|\log\left(\Psi_\nu\left(p\right)\right) - \log\left(\Psi_{\nu_n}\left(p\right)\right)\right| dp \xrightarrow{n \to \infty} 0. \qedhere
\end{equation*}

Let \(\nu \in \mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\}\) and \(\left(\nu_n\right)_{n \in \N}\) be a sequence in \(\mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\}\) with

\[\nu_n \underset{w}{\xrightarrow{n \to \infty}} \nu.\]

\[\int_{\R_+} f \,d(\cT\nu_n) \xrightarrow{n \to \infty} \int_{\R_+} f \,d(\cT\nu) \quad \forall f \in C_c(\R_+,\C).\]

Since by assumption

\[\nu_n([0,\infty]) = \int_{[0,\infty]} \textbf{1} \,d\nu_n \xrightarrow{n \to \infty} \int_{[0,\infty]} \textbf{1} \,d\nu = \nu([0,\infty]),\]

we have that

\[M \coloneqq \sup_{n \in \N} \,\nu_n([0,\infty]) < \infty.\]

Further, by lem:tNuUniformlyCompact, the functions \((t_{\nu_n})_{n \in \N}\) converge uniformly to \(t_\nu\) on compact sets. Therefore, for every function \(f \in C_c(\R_+,\C)\) one has

\[\left\lVert f \cdot \left(t_{\nu_n} - t_\nu\right)\right\rVert_\infty \xrightarrow{n \to \infty} 0.\]

This yields

\begin{align*}
\left|\int_{\R_+} f \cdot \left(t_\nu - t_{\nu_n}\right) \,d\nu_n\right|\leq \left\lVert f \cdot \left(t_\nu - t_{\nu_n}\right)\right\rVert_\infty \cdot \nu_n([0,\infty]) = \left\lVert f \cdot \left(t_\nu - t_{\nu_n}\right)\right\rVert_\infty \cdot M \xrightarrow{n \to \infty} 0.
\end{align*}

Also we have \(f \cdot t_\nu \in C_c(\R_+,\C) \subeq C([0,\infty],\C)\), so

\[\left|\int_{\R_+} f \cdot t_\nu \,d\nu - \int_{\R_+} f \cdot t_\nu \,d\nu_n\right| \xrightarrow{n \to \infty} 0.\]

This yields

\begin{align*}
\left|\int_{\R_+} f \,d(\cT\nu) - \int_{\R_+} f \,d(\cT\nu_n)\right| &= \left|\int_{\R_+} f \cdot t_\nu \,d\nu - \int_{\R_+} f \cdot t_{\nu_n} \,d{\nu_n}\right|
\\&\leq \left|\int_{\R_+} f \cdot t_\nu \,d\nu - \int_{\R_+} f \cdot t_\nu \,d\nu_n\right| + \left|\int_{\R_+} f \cdot \left(t_\nu - t_{\nu_n}\right) \,d\nu_n\right|
\\&\xrightarrow{n \to \infty} 0 + 0 = 0. \qedhere
\end{align*}

One has

\[\{H \in \HanP : \left\lVert H\right\rVert \leq 1\} = \left\{H_{h_{\nu}}: \nu \in \mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\}\right\}.\]

By prop:cTNuIsCarlesonOfHNu we have

\[\left\{H_{h_\nu}: \nu \in \mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\}\right\} \subeq \{H \in \HanP : \left\lVert H\right\rVert \leq 1\}.\]

We now show the other inclusion. So let \(H \in \HanP\) with \(\left\lVert H\right\rVert \leq 1\). Then, by prop:dDenseNormLeq1, we have

\[H \in \overline{\{H_{h_{W(\mu)}}: \mu \in \mathcal{BM}^{\mathrm{fin}}\left(\R_+\right) \setminus \{0\}\}}^{\mathrm{WOT}}.\]

Since \(H^2(\C_+)\) is separable and therefore the weak operator topology restricted to the closed unit ball is metrizable, this implies that there exists a sequence of measures \((\mu_n)_{n \in \N}\) in \(\mathcal{BM}^{\mathrm{fin}}\left(\R_+\right) \setminus \{0\}\) such that

\[H_{h_{W(\mu_n)}} \underset{\mathrm{WOT}}{\xrightarrow{n \to \infty}} H.\]

For \(n \in \N\) we now set

\[\nu_n \coloneqq \frac 1{(W(\mu_n))([0,\infty])} W(\mu_n).\]

Since, by rem:scaleable, we have

\begin{equation*}
\cT\left(\nu_n\right) = \cT\left(W(\mu_n)\right),
\end{equation*}

by prop:cTNuIsCarlesonOfHNu, we get

\[H_{\cT \nu_n} = H_{\cT W(\mu_n)} = H_{h_{W(\mu_n)}} \underset{\mathrm{WOT}}{\xrightarrow{n \to \infty}} H.\]

Further, by construction, we have

\[\nu_n([0,\infty]) = 1 \quad \forall n \in \N.\]

The set

\[\{\nu \in \mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right): \nu([0,\infty]) \leq 1\}\]

is a closed subset of

\[B_1(0) = \{\nu \in \mathcal{BM}^{\mathrm{fin}}_\C\left([0,\infty]\right): |\nu|([0,\infty]) \leq 1\}\]

with respect to the weak-\(*\)-topology on

\[\mathcal{BM}^{\mathrm{fin}}_\C\left([0,\infty]\right) = C([0,\infty],\C)'\]

and therefore sequentially compact by the Banach–Alaoglu Theorem. Therefore there exists a weakly convergent subsequence of \((\nu_n)_{n \in \N}\). By replacing \((\nu_n)_{n \in \N}\) with that subsequence, we can assume that

\[\nu_n \underset{w}{\xrightarrow{n \to \infty}} \nu\]

for some measure \(\nu \in \mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right)\). Then

\[1 = \nu_n([0,\infty]) = \int_{[0,\infty]} \textbf{1} \,d\nu_n \xrightarrow{n \to \infty} \int_{[0,\infty]} \textbf{1} \,d\nu = \nu([0,\infty]),\]

so \(\nu([0,\infty]) = 1\) and therefore \(\nu \neq 0\).
Then, by prop:weakImpliesCcConvergence, we have

\[\int_{\R_+} f \,d(\cT\nu_n) \xrightarrow{n \to \infty} \int_{\R_+} f \,d(\cT\nu) \quad \forall f \in C_c(\R_+,\C).\]

But, since

\[H_{\cT \nu_n} \underset{\mathrm{WOT}}{\xrightarrow{n \to \infty}} H,\]

we also have

\[\int_{\R_+} f \,d(\cT\nu_n) \xrightarrow{n \to \infty} \int_{\R_+} f \,d\mu_H \quad \forall f \in C_c(\R_+,\C)\]

by prop:topologiesEquiv. This implies \(\mu_H = \cT \nu\) and therefore, by prop:cTNuIsCarlesonOfHNu, we have

\[H = H_{\cT \nu} = H_{h_\nu}. \qedhere\]

One has

\[\{\mu \in \CM : \left\lVert H_\mu\right\rVert \leq 1\} = \left\{\cT \nu: \nu \in \mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\}\right\}.\]

By prop:cTNuIsCarlesonOfHNu we have

\[\left\{H_{h_{\nu}}: \nu \in \mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\}\right\} = \left\{H_{\cT \nu}: \nu \in \mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\}\right\}\]

The statement then follows by thm:HankelHasUnimodMeasure and prop:HankelOneToOne.

For the measure \(\nu \in \mathcal{BM}^{\mathrm{fin}}\left(\left[0,\infty\right]\right)\) defined by

\[d\nu(\lambda) = \frac 2{1+\lambda^2} \,d\lambda\]

we get

\begin{align*}
\Psi_\nu\left(p\right) &= \frac{1}{\pi}\int_{[0,\infty]} \frac{1+\lambda^2}{p^2 + \lambda^2} \cdot \frac 2{1+\lambda^2} \,d\lambda = \frac{2}{\pi |p|}\int_{\R_+} \frac{1}{1 + \big(\frac \lambda{|p|}\big)^2} \cdot \frac 1{|p|} \,d\lambda
\\&= \frac{2}{\pi |p|}\int_{\R_+} \frac{1}{1 + x^2} \,dx = \frac{2}{\pi |p|} \cdot [\arctan(x)]_0^\infty  = \frac 1{\left|p\right|}.
\end{align*}

Then the outer function \(F_\nu = \Out\left(\sqrt{\Psi_\nu}\right)\), by lem:OuterExamples, is given by

\[F_\nu(z) = \frac{1+i}{\sqrt{2z}}, \quad z \in \C_+,\]

where, for \(z \in \C_+\), by \(\sqrt{z}\) we denote the square root of \(z\) with \(\mathrm{Re}(\sqrt{z})>0\).
For \(x > 0\) this yields

\[h_\nu(x) = \frac{F_\nu(x)}{RF_\nu(x)} = \frac{\frac{1+i}{\sqrt{2x}}}{\frac{1+i}{\sqrt{-2x}}} = \frac {\sqrt{-x}}{\sqrt{x}} = i\]

and analogously we get \(h_\nu(x) = -i\) for every \(x<0\), so we have

\[h_\nu = i \cdot \sgn.\]

Further, for \(\lambda \in \R_+\), we have

\[F_\nu(i\lambda)^2 = \left(\frac{1+i}{\sqrt{2i\lambda}}\right)^2 = \frac{(1+i)^2}{2i\lambda} = \frac 1\lambda,\]

\[d(\cT\nu)(\lambda) = \left|F_\nu(i\lambda)\right|^{-2} \frac{1+\lambda^2}{\lambda} \,d\nu(\lambda) = \lambda \cdot \frac{1+\lambda^2}{\lambda} \cdot \frac 2{1+\lambda^2} \,d\lambda = 2 \,d\lambda\]

and therefore

\[\cT \nu = 2 \lambda_1.\]

§.§ Uniqueness of the measure

In sec:UniqueSymbol we have seen, that, defining on \(\mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\}\) the equivalence relation

\[\nu \sim \nu' \quad :\Leftrightarrow \quad (\exists c \in \R_+)\, \nu = c \cdot \nu',\]

one has

\[h_\nu = h_{\nu'} \quad \Leftrightarrow \quad \nu \sim \nu'.\]

This of course implies that

\[\nu \sim \nu' \quad \Rightarrow \quad H_{h_\nu} = H_{h_{\nu'}}\]

and therefore we can define the map

\[\left(\mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\}\right) \slash \sim \,\to \{H \in \HanP: \left\lVert H\right\rVert \leq 1\}, \quad [\nu] \mapsto H_{h_\nu}.\]

The goal of this section is to answer the question whether it is possible to prove an analogue of cor:hNuInjektiveSim, i.e. to answer the question whether the above map is injective. The following lemma shows that, in this generality, this is not true:

Let \(\nu \in \mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\}\). Then the following are equivalent:

* \(\cT \nu = 0\).

* \(\nu = s \cdot \delta_0 + t \cdot \delta_\infty\) for some \(s,t \in \R_{\geq 0}\).

That \(\cT \nu = 0\) is equivalent to

\[0 = d(\cT \nu)(\lambda) = \left|F_\nu(i\lambda)\right|^{-2} \frac{1+\lambda^2}{\lambda} \,d\nu(\lambda), \quad \lambda \in \R_+\]

and therefore to

\[\nu(\R_+) = 0,\]

\[\left|F_\nu(i\lambda)\right|^{-2} \frac{1+\lambda^2}{\lambda} > 0 \quad \forall \lambda \in \R_+.\]

This shows that for a measure \(\nu \in \mathcal{BM}^{\mathrm{fin}}\left(\left[0,\infty\right]\right) \setminus \{0\}\) one has

\[\cT \nu = 0 \quad \Leftrightarrow \quad \nu(\R_+)=0 \quad \Leftrightarrow \quad (\exists s,t \in \R_{\geq 0}:  \nu = s \cdot \delta_0 + t \cdot \delta_\infty). \qedhere\]

* For the measure \(\nu = \pi \cdot \delta_0\), one gets

\[\Psi_\nu\left(p\right) = \int_{[0,\infty]} \frac{1+\lambda^2}{p^2 + \lambda^2} \cdot \,d\delta_0(\lambda) = \frac{1+0^2}{p^2 + 0^2} = \frac 1{p^2}.\]

Then, by lem:OuterExamples, one has

\[F_\nu(z) = \frac iz, \quad z \in \C_+\]

and therefore

\[h_\nu(x) = \frac{F_\nu(x)}{RF_\nu(x)} = \frac{\frac ix}{\frac i{-x}} = -1, \quad x \in \R^\times\]

and indeed, one has \(H_{-\textbf{1}} = -p_+ R p_+^* = 0\).

* For the measure \(\nu = \pi \cdot \delta_\infty\), one gets

\[\Psi_\nu\left(p\right) = \int_{[0,\infty]} \frac{1+\lambda^2}{p^2 + \lambda^2} \cdot \,d\delta_\infty(\lambda) = 1.\]

Then, one has \(F_\nu \equiv 1\) and therefore

\[h_\nu = \frac{F_\nu}{RF_\nu} = \frac{\textbf{1}}{R\textbf{1}} = \textbf{1}\]

and indeed, one has \(H_{\textbf{1}} = p_+ R p_+^* = 0\).

This example shows that the map

\[\left(\mathcal{BM}^{\mathrm{fin}}\left([0,\infty]\right) \setminus \{0\}\right) \slash \sim \,\to \{H \in \HanP: \left\lVert H\right\rVert \leq 1\}, \quad [\nu] \mapsto H_{h_\nu}\]

is not injective, since we have \(\pi \cdot \delta_0 \not\sim \pi \cdot \delta_\infty\) but

\[H_{h_{\pi \cdot \delta_0}} = 0 = H_{h_{\pi \cdot \delta_\infty}}.\]
However, we will now see that, when one restricts it to positive Hankel operators with a non-zero fixed point, it is in fact a bijection:

The map

\begin{align*}
W\left(\mathcal{BM}^{\mathrm{fin}}\left(\R_+\right) \setminus \{0\}\right) \slash \sim \,&\to \{H \in \HanP: \left\lVert H\right\rVert \leq 1, \,\ker(H - \textbf{1}) \neq \{0\}\}
\\ [W(\mu)] &\mapsto H_{h_{W(\mu)}}
\end{align*}

is a bijection.

The surjectivity follows immediately by prop:HankelFixedPoint. We now show the injectivity. So let \(\mu,\mu' \in \mathcal{BM}^{\mathrm{fin}}\left(\R_+\right) \setminus \{0\}\) with

\[H_{h_{W(\mu)}} = H_{h_{W(\mu')}}.\]

We set

\[f \coloneqq F_{W(\mu)} \in \ker\left(\theta_{h_{W(\mu)}}-{\bf 1}\right) \cap \Out^2(\C_+)\]

\[g \coloneqq F_{W(\mu')} \in \ker\left(\theta_{h_{W(\mu')}}-{\bf 1}\right) \cap \Out^2(\C_+)\]

(cf. def:FNuHNu). Then, for \(t \geq 0\), we have

\begin{align*}
\braket*{S_t f}{\left(\theta_{h_{W(\mu)}}-{\bf 1}\right)g} &= \braket*{S_t f}{(H_{h_{W(\mu)}}-\textbf{1}) g}
\\&= \braket*{S_t f}{(H_{h_{W(\mu')}}-\textbf{1}) g} = \braket*{S_t f}{\left(\theta_{h_{W(\mu')}}-{\bf 1}\right)g} = 0
\end{align*}

and for \(t \leq 0\) we have

\begin{align*}
\braket*{S_t f}{\theta_{h_{W(\mu)}} g} &= \braket*{\theta_{h_{W(\mu)}} S_t f}{ g} = \braket*{S_{-t} \theta_{h_{W(\mu)}} f}{g} = \braket*{S_{-t} f}{g}
\\&= \braket*{S_{-t} f}{\theta_{h_{W(\mu')}} g} = \braket*{S_{-t} f}{H_{h_{W(\mu')}} g} = \braket*{S_{-t} f}{H_{h_{W(\mu)}} g}
\\&= \braket*{S_{-t} f}{\theta_{h_{W(\mu)}} g} = \braket*{\theta_{h_{W(\mu)}} S_{-t} f}{g} = \braket*{S_t \theta_{h_{W(\mu)}} f}{g} = \braket*{S_t f}{g}
\end{align*}

and therefore

\begin{align*}
\braket*{S_t f}{\left(\theta_{h_{W(\mu)}}-{\bf 1}\right)g} &= \braket*{S_t f}{\theta_{h_{W(\mu)}} g} - \braket*{S_t f}{g} = 0.
\end{align*}

This yields

\[\braket*{S_t f}{\left(\theta_{h_{W(\mu)}}-{\bf 1}\right)g} = 0 \quad \forall t \in \R.\]

Since, by cor:OuterDense, we have \(\overline{\spann S(\R)f} = L^2(\R,\C)\), we get \(\left(\theta_{h_{W(\mu)}}-{\bf 1}\right)g = 0\). This implies \(\theta_{h_{W(\mu)}} g = g\) and therefore

\[h_{W(\mu)} = \frac g{Rg} = \frac{F_{W(\mu')}}{RF_{W(\mu')}} = h_{W(\mu')},\]

so, by cor:hNuInjektiveSim, we get \([W(\mu)] = [W(\mu')]\).

§ APPLICATION TO STANDARD SUBSPACES

In this chapter we will investigate so called standard subspaces and certain unitary one-parameter groups that are compatible with a standard subspace. We will relate these unitary one-parameter groups to real (reflection positive) ROPGs using Borchers' Theorem. To understand Borchers' Theorem we need some basic theory of standard subspaces as well as basics of the representation theory of the affine group \(\mathrm{Aff}(\R)\).

§.§ Standard subspaces

We start with the definition and basic theory of standard subspaces:
(cf. <cit.>)
Let \(\cH\) be a complex Hilbert space.

* A closed real subspace \(\gls*{V} \subeq \cH\) is called a standard subspace, if

\[\sV \cap i\sV = \{0\} \qquad \text{and} \qquad \oline{\sV + i\sV} = \cH.\]

* Given a standard subspace \(\sV\) we consider the densely defined Tomita operator given by

\[\gls*{TV} : \sV+i\sV \to \cH, \quad x+iy \mapsto x-iy.\]

* We write \(\gls*{JV}\) for the anti-unitary involution and \(\gls*{DeltaV}\) for the strictly positive, densely defined operator one gets from the polar decomposition

\[T_\sV = J_\sV \Delta_\sV^{\frac 12}\]

of the Tomita operator. We call \((\Delta_\sV,J_\sV)\) the pair of modular objects of \(\sV\).

(cf. <cit.>)
Let \(\cH\) be a complex Hilbert space and \(\sV \subeq \cH\) be a standard subspace. Then the pair \((\Delta_\sV,J_\sV)\) of modular objects of \(\sV\) satisfies the relation

\[J_V \Delta_V = \Delta_V^{-1}J_V\]

and one has

\[\sV = \mathrm{Fix}\big(J_\sV \Delta_\sV^{\frac 12}\big)\]

Conversely, for every pair of a strictly positive selfadjoint operator \(\Delta\) and an anti-unitary involution \(J\) satisfying \(J\Delta = \Delta^{-1}J\), the space \(\sV \coloneqq \mathrm{Fix}\big(J \Delta^{\frac 12}\big)\) is a standard subspace.

This proposition shows that, starting with a standard subspace \(\sV\) and considering its modular pair \((\Delta_\sV,J_\sV)\), one gets another modular pair by \((\Delta_\sV^{-1},J_\sV)\), which gives rise to another standard subspace. To get a more explicit form of that standard subspace, we make the following definition:

Let \(\cH\) be a Hilbert space and \(\sV \subeq \cH\) be a standard subspace. Then we set

\[\gls*{Vprime} \coloneqq (i\sV)^{\perp_\R},\]

where by \(\perp_\R\) we denote the orthogonal complement with respect to the real scalar product

\[\braket*{\cdot}{\cdot}_\R: \cH \times \cH \to \R, \quad (v,w) \mapsto \Re \,\braket*{v}{w}.\]

(cf. <cit.>)
Let \(\cH\) be a Hilbert space and \(\sV \subeq \cH\) a standard subspace. Then \(\sV'\) is also a standard subspace and

\[J_{\sV'} = J_{\sV} \qquad \text{and} \qquad \Delta_{\sV'} = \Delta_{\sV}^{-1}.\]

§.§ Positive energy representations of the affine group

In this section we provide some basic results of the representation theory of the affine group \(\mathrm{Aff}(\R)\). The affine group is an example of a so called graded group, defined as follows:

A graded group is a pair \(\left(G,\epsilon_G\right)\) of a group \(G\) and a group homomorphism \(\epsilon_G:G \to \left\lbrace \pm 1 \right\rbrace \). We set \(G_{\pm}\coloneqq \epsilon_G^{-1}\left(\left\lbrace\pm 1\right\rbrace\right)\).

* The affine group \(\gls*{Aff} = \R \rtimes \R^\times\) becomes a graded group with the grading

\[\gls*{Aff+} = \R \rtimes \R_+ \quad \text{and} \quad \mathrm{Aff}(\R)_- = \R \rtimes \R_-.\]

* Given a complex Hilbert space \(\cH\) we consider the group \(\gls*{AUH}\) of unitary and anti-unitary operators on \(\cH\). This becomes a graded group with the grading

\[\mathrm{AU}\left(\cH\right)_+ = \U(\cH) \quad \text{and} \quad \mathrm{AU}\left(\cH\right)_- = \mathrm{AU}\left(\cH\right) \setminus \U(\cH).\]

Now that we have the concept of a graded group, we want to define what a representation of a graded group is:

An anti-unitary representation of a graded group \((G,\epsilon_G)\) on a Hilbert space \(\cH\) is a morphism of graded groups \({\rho:G \to \mathrm{AU}\left(\cH\right)}\).

We are especially interested in so called positive energy representations of the affine group, which are defined as follows:

* Given a unitary representation \({\rho:\mathrm{Aff}(\R)_+ \to \mathrm{U}\left(\cH\right)}\) of the \(ax+b\)-group \(\mathrm{Aff}(\R)_+\), we say that \(\rho\) is a positive energy representation, if, for the one-parameter group

\[U: \R \to \U(\cH), \quad t \mapsto \rho(t,1),\]

one has \(\partial U \geq 0\). We call the representation \(\rho\) a strictly positive energy representation if additionally \({\ker (\partial U) = \{0\}}\).

* An anti-unitary representation \({\rho:\mathrm{Aff}(\R) \to \mathrm{AU}\left(\cH\right)}\) of the affine group is a (strictly) positive energy representation if the restriction \({\rho\big|_{\mathrm{Aff}(\R)_+}:\mathrm{Aff}(\R)_+ \to \mathrm{AU}\left(\cH\right)}\) is a (strictly) positive energy representation.

Let \(\cK\) be a real Hilbert space. We set

\[\gls*{HK} \coloneqq L^2(\R,\cK_\C)^\sharp.\]

Then \(\cH_\cK\) becomes a complex Hilbert space with the complex structure

\[I: \cH_\cK \to \cH_\cK, \quad f \mapsto i \cdot \sgn \cdot f.\]

Then the map \(\gls*{rhoK}: \mathrm{Aff}(\R) \to \mathrm{AU}(\cH_\cK)\) with

\[(\rho_\cK(t,1)f)(x) \coloneqq e^{itx} f(x), \quad (\rho_\cK(0,e^s)f)(x) \coloneqq e^{\frac s2}f(e^s x) \qquad s,t \in \R\]

\[(\rho_\cK(0,-1)f)(x) \coloneqq i \cdot \sgn(x) \cdot f(-x)\]

defines a strictly positive energy representation of the affine group \(\mathrm{Aff}(\R)\).

The following proposition shows that in fact every strictly positive energy representation of \(\mathrm{Aff}(\R)\) is of this form:

Let \((\rho,\cH)\) be a strictly positive energy representation of the affine group \(\mathrm{Aff}(\R)\). Then there exists a real Hilbert space \(\cK\) such that \((\rho,\cH)\) is equivalent to the anti-unitary representation \((\rho_\cK,\cH_\cK)\). Further, the representation \((\rho_\cK,\cH_\cK)\) is irreducible, if and only if \(\cK = \R\).

By <cit.> there exists – up to equivalence – exactly one irreducible unitary representation of \(\mathrm{Aff}(\R)_+\) with strictly positive energy. By <cit.> this representation \(U\) can be realized on the Hilbert space \(L^2(\R_+,\C)\) by

\[(U(t,1)f)(x) = e^{itx} f(x) \quad \text{and} \quad (U(0,e^s)f)(x) \coloneqq e^{\frac s2}f(e^s x), \qquad f \in L^2(\R_+,\C), s,t,x \in \R.\]

Via the map

\[\Psi: L^2\left(\R_+,\C\right) \to L^2\left(\R,\C\right)^\sharp, \quad (\Psi f)(x) \coloneqq \begin{cases} f(x) & \text{if } x > 0 \\[0.75ex] \overline{f(-x)} & \text{if } x < 0,\end{cases}\]

this representation is equivalent to the representation \(\rho_\R\big|_{\mathrm{Aff}(\R)_+}\).

Further, by <cit.>, every unitary representation of \(\mathrm{Aff}(\R)_+\) with strictly positive energy is a multiple of this unique irreducible representation, so every strictly positive energy representation \((\rho,\cH)\) of the group \(\mathrm{Aff}(\R)_+\) is equivalent to

\[\tilde \rho_\cK \coloneqq \rho_\cK\big|_{\mathrm{Aff}(\R)_+}: \mathrm{Aff}(\R)_+ \to \U(\cH)\]

for some real Hilbert space \(\cK\). Now, by <cit.>, for every unitary representation of \(\mathrm{Aff}(\R)_+\), there exists an – up to equivalence – unique extension to an anti-unitary representation of \(\mathrm{Aff}(\R)\) on the same Hilbert space. Since \(\rho_\cK\) is such an extension of \(\tilde \rho_\cK\), every anti-unitary representation of \(\mathrm{Aff}(\R)\) is equivalent to \(\rho_\cK\) for some real Hilbert space \(\cK\). Finally, by  the representation \(\rho_\cK\) is irreducible, if and only if \(\tilde \rho_\cK\) is irreducible, which is the case, if and only if \(\cK = \R\).

§.§ Extending Borchers' Theorem beyond standard pairs

To state Borchers' Theorem, that links strictly positive energy representations of the affine group to standard subspaces, we need one more definition:

Let \(\cH\) be a Hilbert space, \(\sV \subeq \cH\) a standard subspace and \(U: \R \to \U(\cH)\) a strongly continuous one-parameter group. We call the pair \((\sV,U)\) a standard pair on \(\cH\), if

\[U_t \sV \subeq \sV \quad \forall t \in \R_+ \qquad \text{and} \qquad \partial U \geq 0.\]

We say that a standard pair \((\sV,U)\) is non-degenerate if \(\ker (\partial U) = \{0\}\).

(Borchers' Theorem (one-particle))(<cit.>, <cit.>)
Let \(\cH\) be a Hilbert space and let \((\sV,U)\) be a non-degenerate standard pair on \(\cH\). Then the following commutation relations hold:

\[\Delta_\sV^{is} U_t \Delta_\sV^{-is} = U_{e^{-2\pi s} t}, \qquad J_\sV U_t J_\sV = U_{-t} \qquad \forall s,t \in \R.\]

\[\rho(t,1) \coloneqq U_t, \quad \rho(0,e^s) \coloneqq \Delta_\sV^{-\frac{is}{2\pi}} \quad \text{and} \quad \rho(0,-1) \coloneqq J_\sV \qquad s,t \in \R\]

defines a strictly positive energy representation \(V\) of \(\mathrm{Aff}(\R)\) on \(\cH\).

Borchers' Theorem shows us that for every non-degenerate standard pair we get a strictly positive energy representation of the affine group \(\mathrm{Aff}(\R)\). The following proposition shows that conversely, starting with a strictly positive energy representation of the affine group, we also obtain a non-degenerate standard pair:

For every real Hilbert space \(\cK\), the space \(\gls*{VK} \coloneqq H^2(\C_+,\cK_\C)^\sharp\) is a standard subspace in the Hilbert space \(\cH_\cK\) with

\[J_{\sV_\cK} = \rho_\cK(0,-1) \qquad \text{and} \qquad \Delta_{\sV_\cK}^{-\frac{is}{2\pi}} = \rho_\cK(0,e^s), \quad s \in \R.\]

Further, defining the one-parameter group

\[U_\cK: \R \to \U(\cH), \quad t \mapsto \rho_\cK(t,1),\]

the pair \((\sV_\cK,U_\cK)\) is a non-degenerate standard pair.

We consider the set

\[\cS_\pi \coloneqq \{z \in \C: 0 < \Im(z) < \pi\}\]

and the corresponding Hardy space

\[H^2(\cS_\pi) \coloneqq \left\{f \in \cO(\cS_\pi): \sup_{y \in (0,\pi)} \int_\R |f(x+iy)|^2 \,dx < \infty \right\}.\]

By <cit.> for every function \(f \in H^2(\cS_\pi)\) the lower and upper boundary values

\[f(x) \coloneqq \lim_{\epsilon \downarrow 0} f(x+i\epsilon) \quad \text{and} \quad f(x+i\pi) \coloneqq \lim_{\epsilon \downarrow 0} f(x+i(\pi-\epsilon))\]

exist for almost every \(x \in \R\) and the lower boundary values define a function in \(L^2(\R,\C)\). Now, by <cit.>, the space

\[H \coloneqq \{f \in H^2(\cS_\pi): \overline{f(x+i\pi)} = f(x) \text{ for almost every }x \in \R\}\]

is a standard subspace in \(L^2(\R,\C)\) with

\[J_H f = \overline{f} \qquad \text{and} \qquad \left(\Delta_H^{-\frac{is}{2\pi}}f\right)(x) = f(x+s), \quad f \in L^2(\R,\C), s,x \in \R.\]

We now consider the unitary map

\[\Psi: L^2(\R,\C) \to L^2(\R,\C)^\sharp = \cH_\R, \quad \begin{cases} \frac {(1+i)}{2\sqrt{x}} f(\log(x)) & \text{if } x > 0 \\[1.5ex] \frac {(1+i)}{2\sqrt{-x}} \overline{f(\log(-x))} & \text{if } x < 0.\end{cases}\]

\[\Psi(H) = H^2(\C_+)^\sharp = \sV_\R.\]

This shows that \(\sV_\R\) is a standard subspace in the Hilbert space \(\cH_\R\) with

\[J_{\sV_\R} = \Psi \circ J_H \circ \Psi^{-1} = \rho_\R(0,-1)\]

\[\Delta_{\sV_\R}^{-\frac{is}{2\pi}} = \Psi \circ \Delta_H^{-\frac{is}{2\pi}} \circ \Psi^{-1} = \rho_\R(0,e^s), \quad s \in \R.\]

The first statement then follows by taking direct sums. The second statement follows immediately by the fact that

\[\left(U_\cK\right)_t H^2(\C_+,\cK_\C)^\sharp = S_t H^2(\C_+,\cK_\C)^\sharp \subeq H^2(\C_+,\cK_\C)^\sharp \quad \forall t \in \R_+\]

(cf. thm:LaxPhillipsReal) and that \(\rho_\cK\) is a strictly positive energy representation.

Let \(\cH\) be a Hilbert space and let \((\sV,U)\) be a non-degenerate standard pair on \(\cH\). Then there exists a real Hilbert space \(\cK\) and a unitary operator \(\psi: \cH \to \cH_\cK\) such that

\[\psi(\sV) = \sV_\cK, \qquad \psi \circ J_\sV \circ \psi^{-1} = \theta_{i \cdot \sgn} \qquad \text{and} \qquad \psi \circ U_t \circ \psi^{-1} = S_t \quad \forall t \in \R\]

(cf. def:defTheta, thm:LaxPhillipsReal).

By prop:RepAffineForm there exists a real Hilbert space \(\cK\) and a unitary operator \(\psi: \cH \to \cH_\cK\) such that

\[\psi \circ J_\sV \circ \psi^{-1} = \rho_\cK(0,-1) = \theta_{i \cdot \sgn} \qquad \text{and} \qquad \psi \circ U_t \circ \psi^{-1} = \rho_\cK(t,1) = S_t \quad \forall t \in \R\]

\[\psi \circ \Delta_\sV^{-\frac{is}{2\pi}} \circ \psi^{-1} = \rho_\cK(0,e^s) \quad \forall s \in \R.\]

This, by prop:WhatIsStandard, implies that \(\psi(\sV)\) and \(\sV_\cK\) are both standard subspaces in \(\cH_\cK\) with

\[J_{\psi(\sV)} = J_{\sV_\cK} \qquad \text{and} \qquad \Delta_{\psi(\sV)} = \Delta_{\sV_\cK}\]

and therefore we have

\[\psi(\sV) = \Fix\left(J_{\psi(\sV)}\Delta_{\psi(\sV)}^{\frac 12}\right) = \Fix\left(J_{\sV_\cK}\Delta_{\sV_\cK}^{\frac 12}\right) = \sV_\cK. \qedhere\]

This corollary provides a normal form for non-degenerate standard pairs. The goal of this section is to provide similar normal form theorems for pairs \((V,U)\) of a standard subspace \(\sV\) and a unitary one-parameter group \(U\) on \(\cH\) with \(U_t \sV \subeq \sV\) for all \(t \in \R_+\), but without the additional assumption that the infinitesimal generator of \(U\) is positive. We want to replace this assumption by the weaker assumption that – as it was the case for non-degenerate standard pairs by cor:NormalformStandardPair – there exists an isomorphism between \((\cH,\sV,U)\) and \((\cH_\cK,\sV_\cK,S)\) for some real Hilbert space \(\cK\). This condition, by the Lax–Phillips Theorem (thm:LaxPhillipsReal), is equivalent to \((\cH,\sV,U)\) being a real ROPG. This motivates the following definition:

Let \(\cH\) be a complex Hilbert space, \(\sV \subeq \cH\) a standard subspace and \(U: \R \to \U(\cH)\) a strongly continuous one-parameter group.

* We call \((\sV,U)\) a regular pair, if \((\cH,\sV,U)\) is a real ROPG.

* We call \((\sV,U)\) a reflection positive regular pair, if \((\cH,\sV,U,J_\sV)\) is a real reflection positive ROPG.

It follows directly from the definition that every reflection positive regular pair \((\sV,U)\) is in particular a regular pair.

First, we prove that both of these conditions are fulfilled if \((\sV,U)\) is a non-degenerate standard pair, to show that we are in fact dealing with a generalization of non-degenerate standard pairs. For this we need the following proposition:

Let \(\cH\) be a complex Hilbert space and \(\sV \subeq \cH\)  be a standard subspace. Then \((\cH,\sV,J_\sV)\) is a maximal real RPHS.

Given a standard subspace \(\sV \subeq \cH\), by prop:StandardSubspaceFixed, for every \(v \in \sV \setminus \{0\}\), one has

\[J_\sV \Delta_\sV^{\frac 12}v = v\]

and therefore \(\Delta_\sV^{\frac 12}v \neq 0\), so

\[\braket*{v}{J_\sV v} = \braket*{v}{J_\sV \big(J_\sV \Delta_\sV^{\frac 12}\big)v} = \braket*{v}{\Delta_\sV^{\frac 12} v} > 0,\]

hence the triple \((\cH,\sV,J_\sV)\) is a RPHS. By the same argument, using that \(J_\sV = J_{\sV'}\) by lem:JSymplCompl, we have that

\[\braket*{v}{J_\sV v} = \braket*{v}{J_{\sV'} v} > 0\]

for every \(v \in \sV' \setminus \{0\}\). This yields

\[\braket*{iv}{J_\sV iv} = \braket*{iv}{-iJ_\sV v} = -\braket*{v}{J_\sV v} < 0\]

for every \(v \in \sV' \setminus \{0\}\). Since

\[i\sV' = \sV^{\perp_\R},\]

this yields that

\[\braket*{v}{J_\sV v} < 0 \quad \forall v \in \sV^{\perp_\R}.\]

We now assume that \(w \in \cH\) such that \((\cH,\sV+\R w,J_\sV)\) is a real RPHS. Then \(\sV+\R w = \sV+\R P_{\sV^{\perp_\R}} w\), so we can assume that \(w \in \sV^{\perp_\R}\), which by the above implies that \(w = 0\), so \((\cH,\sV,J_\sV)\) is a maximal RPHS.

Let \(\cH\) be a complex Hilbert space, \(\sV \subeq \cH\) a standard subspace and \(U: \R \to \U(\cH)\) a strongly continuous one-parameter group. If \((\sV,U)\) is a non-degenerate standard pair, then \((\sV,U)\) is a reflection positive regular pair.

By prop:StandardSubspaceMaxPos the triple \((\cH,\sV,J_\sV)\) is a RPHS and by Borchers' Theorem (thm:Borchers) we have \(J_\sV \circ U_t = U_{-t} \circ J_\sV\). Together with the assumption

\[U_t \sV \subeq \sV, \quad \forall t \in \R_+,\]

this implies that \((\cH,\sV,U,J_\sV)\) is a reflection positive unitary one-parameter group. Further, by cor:NormalformStandardPair there exists a real Hilbert space \(\cK\) such that the triple \((\cH,\sV,U)\) is isomorphic to \((\cH_\cK,\sV_\cK,S)\), which is a real ROPG by the Lax–Phillips Theorem (thm:LaxPhillipsReal). This implies that \((\cH,\sV,U,J_\sV)\) is a reflection positive ROPG.

By the Lax–Phillips Theorem (thm:LaxPhillipsReal) we know that, for every regular pair \((\sV,U)\) on a Hilbert space \(\cH\), the ROPG \((\cH,\sV,U)\) is equivalent to \((\cH_\cK,\sV_\cK,S)\) for some real Hilbert space \(\cK\). We now want to see which ROPGs \((\cH,\sV,U)\) correspond to the multiplicity free case \(\cK = \R\). For this we make the following definition:

Let \(\cH\) be a complex Hilbert space and let \(U: \R \to \U(\cH)\) be a strongly continuous one-parameter group. We say that \(U\) is \(\R\)-cyclic, if there exists \(v \in \cH\) such that

\[\overline{\spann_\R U(\R)v} = \cH.\]

Let \(\cH\) be a complex Hilbert space and \((\sV,U)\) be a regular pair on \(\cH\). Then the multiplicity space of \((\cH,\sV,U)\) is isomorphic to \(\R\), if and only if \(U\) is \(\R\)-cyclic.

If the multiplicity space of \((\cH,\sV,U)\) is isomorphic to \(\R\), by the Lax–Phillips Theorem (thm:LaxPhillipsReal), we can without loss of generality assume that \((\cH,\sV,U) = (L^2(\R,\C)^\sharp,H^2(\C_+)^\sharp,S)\). Then, for the function

\[F(z) \coloneqq \frac{iz}{(z+i)^2}, \quad z \in \C_+,\]

by lem:OuterExamples, lem:OuterSymmetric and thm:OuterBetrag, we have \(F \in \Out^2(\C_+)^\sharp\). This, by cor:OuterDense, yields

\[L^2(\R,\C)^\sharp = \overline{\spann_\R S(\R)F}\]

and therefore \(S\) is \(\R\)-cyclic.

Now, for the converse, we assume that \((\cH,\sV,U)\) is a real ROPG such that \(U\) is \(\R\)-cyclic. Then, by the Lax–Phillips Theorem (thm:LaxPhillipsReal), we can without loss of generality assume that \((\cH,\sV,U) = (L^2(\R,\cK_\C)^\sharp,H^2(\C_+,\cK_\C)^\sharp,S)\) for some real Hilbert space \(\cK\). The \(\R\)-cyclicity then implies that there exists \(F \in L^2(\R,\cK_\C)^\sharp\) such that

\[L^2(\R,\cK_\C)^\sharp = \overline{\spann_\R S(\R)F}.\]

This in particular implies that, for \(v \in \cK \setminus \{0\}\), setting \(F_v \coloneqq \braket*{v}{F(\cdot)} \in L^2(\R,\C)^\sharp\), we have

\[\overline{\spann_\R S(\R)F_v} \otimes v = \overline{\spann_\R S(\R)P_{\C v}F} = P_{\C v} \,\overline{\spann_\R S(\R)F} = P_{\C v} L^2(\R,\cK_\C)^\sharp = L^2(\R,\C)^\sharp \otimes v\]

and therefore

\[\overline{\spann_\R S(\R)F_v} = L^2(\R,\C)^\sharp,\]

so, by Wiener's Tauberian Theorem (<cit.>), we have

\[0 \neq F_v(x) = \braket*{v}{F(x)}\]

for almost every \(x \in \R\). We now define the function \(f \in L^2(\R,\C)^\sharp\) by

\[f(x) = \left\lVert F(x)\right\rVert, \quad x \in \R.\]

Then also \(f(x) \neq 0\) for almost every \(x \in \R\), so, by Wiener's Tauberian Theorem (<cit.>), we have

\[\overline{\spann_\R S(\R)f} = L^2(\R,\C)^\sharp.\]

We now consider the following densely defined linear operator

\[\phi: L^2(\R,\cK_\C)^\sharp \supseteq L^\infty(\R,\C)^\sharp \cdot F \to L^2(\R,\C)^\sharp, \quad g \cdot F \mapsto g \cdot f.\]

For \(g \in L^\infty(\R,\C)^\sharp\) we have

\[\left\lVert \phi(g \cdot F)\right\rVert^2 = \left\lVert g \cdot f\right\rVert^2 = \int_\R |g(x)|^2 \cdot |f(x)|^2 \,dx = \int_\R |g(x)|^2 \cdot \left\lVert F(x)\right\rVert^2 \,dx = \left\lVert g \cdot F\right\rVert^2,\]

so \(\phi\) is isometric and therefore we can extend it to a unitary operator

\[\tilde \phi: L^2(\R,\cK_\C)^\sharp \to \overline{\spann_\R S(\R)f} = L^2(\R,\C)^\sharp.\]

By construction we have

\[\phi \circ S_t = S_t \circ \phi \quad \forall t \in \R\]

and therefore also

\[\tilde \phi \circ S_t = S_t \circ \tilde \phi \quad \forall t \in \R.\]

So, setting

\[\cE_+ \coloneqq \tilde \phi(H^2(\C_+,\cK_\C)^\sharp),\]

the operator \(\tilde \phi\) is an isomorphism between the ROPG \((\cH,\sV,U) = (L^2(\R,\cK_\C)^\sharp,H^2(\C_+,\cK_\C)^\sharp,S)\) and the ROPG \((L^2(\R,\C)^\sharp,\cE_+,S)\). By thm:IndependentFromSubspace the latter is equivalent to the real ROPG \((L^2(\R,\C)^\sharp,H^2(\C_+)^\sharp,S)\). Since, by thm:kernelAdjointGeneralReal, the multiplicity space is unique up to isomorphism, this yields that \(\cK = \R\).

We now summarize our results in the following three theorems that give a normal form for regular pairs, reflection positive regular pairs and non-degenerate standard pairs respectively:

Let \(\cH\) be a complex Hilbert space and \((\sV,U)\) be a regular pair on \(\cH\). Then the following hold:

* There exists a real Hilbert space \(\cK\) such that the real ROPG \((\cH,\sV,U)\) is equivalent to \((\cH_\cK,\sV_\cK,S)\).

* Denoting by \(\phi\) the isomorphism between \((\cH,\sV,U)\) and \((\cH_\cK,\sV_\cK,S)\) and setting

\[\theta \coloneqq \phi \circ J_\sV \circ \phi^{-1} \in B(\cH_\cK),\]

the triple \((\cH_\cK,\sV_\cK,\theta)\) is a maximal RPHS.

* One has \(\cK = \R\), if and only if \(U\) is \(\R\)-cyclic.

Statement (a) follows immediately by the Lax–Phillips Theorem (thm:LaxPhillipsReal), (b) follows by prop:StandardSubspaceMaxPos and (c) by prop:MultiplicityRCyclic.

Let \(\cH\) be a complex Hilbert space and \((\sV,U)\) be a reflection positive regular pair on \(\cH\). Then the following hold:

* There exists a real Hilbert space \(\cK\) and a function \(h \in L^\infty(\R,\U(\cK_\C))^\flat\) with \(h = h^\sharp\) such that the real reflection positive ROPG \((\cH,\sV,U,J)\) is equivalent to \((\cH_\cK,\sV_\cK,S,\theta_h)\).

* The triple \((\cH_\cK,\sV_\cK,\theta_h)\) is a maximal RPHS.

* One has \(\cK = \R\), if and only if \(U\) is \(\R\)-cyclic. In this case there exists an – up to multiplication with a positive constant – unique measure \({\nu \in \mathcal{BM}^{\mathrm{fin}}([0,\infty]) \setminus \{0\}}\) such that \(h = h_\nu\).

Statement (a) follows immediately by thm:NormalformRPROPG and (b) follows by prop:StandardSubspaceMaxPos. We now prove (c). By prop:MultiplicityRCyclic we have that \(\cK = \R\), if and only if \(U\) is \(\R\)-cyclic. By (b) we have that \((\cH_\cK,\sV_\cK,\theta_h)\) is a maximal real RPHS, which by cor:EPlusMaximalReal is equivalent to \((L^2(\R,\C),H^2(\C_+),S)\) being a maximal complex RPHS. By thm:GeneralizedMaxPosForm this in turn is equivalent to the existence of a measure \({\nu \in \mathcal{BM}^{\mathrm{fin}}([0,\infty]) \setminus \{0\}}\) such that \(h = h_\nu\). The uniqueness assertion about this measure then follows by cor:hNuInjektiveSim.

Let \(\cH\) be a complex Hilbert space and \((\sV,U)\) be a non-degenerate standard pair on \(\cH\). Then the following hold:

* There exists a real Hilbert space \(\cK\) such that the real reflection positive ROPG \((\cH,\sV,U,J)\) is equivalent to \((\cH_\cK,\sV_\cK,S,\theta_{i \cdot \sgn})\).

* The triple \((\cH_\cK,\sV_\cK,\theta_{i \cdot \sgn})\) is a maximal RPHS.

* One has \(\cK = \R\), if and only if \(U\) is \(\R\)-cyclic. In this case one has \(\theta_{i \cdot \sgn} = \theta_\nu\) for the measure \(\nu \in \mathcal{BM}^{\mathrm{fin}}([0,\infty]) \setminus \{0\}\) given by

\[d\nu(\lambda) = \frac 2{1+\lambda^2}, \quad \lambda \in [0,\infty].\]

Statement (a) follows immediately by cor:NormalformStandardPair, (b) follows by prop:StandardSubspaceMaxPos and (c) by prop:MultiplicityRCyclic and ex:Lebesgue.

The results of these three theorems are presented in the following table:

30ptM\((\sV,U)\)-30ptM     regular pair     \(\substack{\text{\normalsize \bf reflection positive} \\ \text{\normalsize \bf regular pair}}\)     \(\substack{\text{\normalsize \bf non-degenerate} \\ \text{\normalsize \bf standard pair}}\)

30ptM Group \(G\)-30ptM     \(\R\)     \(\R \rtimes \{-1,1\}\)     \(\mathrm{Aff}(\R) = \R \rtimes \R^\times\)

30ptM\(\substack{\text{\normalsize \bf Representation} \\ \text{\normalsize \boldmath\(\rho: G \to \mathrm{AU}(\cH)\)}}\)-30ptM      \(\rho(t) = U_t\)     \(\substack{\text{\normalsize \(\rho(t,1) = U_t\)} \\ \text{\normalsize \(\rho(0,-1) = J_\sV\)} \\ \text{\normalsize (\fref{def:RPHS})}}\)     \(\substack{\text{\normalsize \(\rho(t,1) = U_t\)} \\ \text{\normalsize \(\rho(0,e^s) = \Delta^{-\frac{is}{2\pi}}\)} \\ \text{\normalsize \(\rho(0,-1) = J_\sV\)} \\ \text{\normalsize (\fref{thm:Borchers})}}\)

30ptM\(\substack{\text{\normalsize \bf Normal form} \\ \text{\normalsize \bf for \boldmath\((\cH,\sV,U,J_\sV)\)}}\) -30ptM     \(\substack{\text{\normalsize \((\cH_\cK,\sV_\cK,S,\theta)\) with} \\ \text{\normalsize \((\cH_\cK,\sV_\cK,\theta)\) being a} \\ \text{\normalsize maximal RPHS} \\ \text{\normalsize (\fref{thm:StandardSubspaceROPG})}}\)     \(\substack{\text{\normalsize \((\cH_\cK,\sV_\cK,S,\theta_h)\) with} \\ \text{\normalsize \(h = h^\sharp = h^\flat\) and}\\ \text{\normalsize \((\cH_\cK,\sV_\cK,\theta_h)\) being a} \\ \text{\normalsize maximal RPHS} \\ \text{\normalsize (\fref{thm:StandardSubspaceRPROPG})}}\)     \(\substack{\text{\normalsize \((\cH_\cK,\sV_\cK,S,\theta_{i \cdot \sgn})\)} \\ \text{\normalsize (\fref{thm:StandardSubspaceSP})}}\)

30ptM\(\substack{\text{\normalsize \bf Normal form} \\ \text{\normalsize \bf for \boldmath\((\cH,\sV,U,J_\sV)\)}\\ \text{\normalsize \bf if \boldmath\(U\) is \boldmath\(\R\)-cyclic}}\) -30ptM     \(\substack{\text{\normalsize \((\cH_\R,\sV_\R,S,\theta)\) with} \\ \text{\normalsize \((\cH_\R,\sV_\R,\theta)\) being a} \\ \text{\normalsize maximal RPHS} \\ \text{\normalsize (\fref{thm:StandardSubspaceROPG})}}\)     \(\substack{\text{\normalsize \((\cH_\R,\sV_\R,S,\theta_{h_\nu})\) with} \\ \text{\normalsize \(\nu \in \mathcal{BM}^{\mathrm{fin}}([0,\infty]) \setminus \{0\}\)} \\ \text{\normalsize (\fref{thm:StandardSubspaceRPROPG})}}\)     \(\substack{\text{\normalsize \((\cH_\R,\sV_\R,S,\theta_{i \cdot \sgn}) =\)} \\ \text{\normalsize \((\cH_\R,\sV_\R,S,\theta_\nu)\) with} \\ \text{\normalsize \(d\nu(\lambda) = \frac 2{1+\lambda^2}\),} \\ \text{\normalsize (\fref{thm:StandardSubspaceSP})}}\)

30ptM\(\substack{\text{\normalsize \bf OS transform} \\ \text{\normalsize \bf of \boldmath\((\cH,\sV,J_\sV)\)}\\ \text{\normalsize \bf if \boldmath\(U\) is \boldmath\(\R\)-cyclic}}\)-30ptM     \(\substack{\text{\normalsize\(\overline{\sV_\R \slash \cN_\theta}\)} \\ \text{\normalsize (\fref{def:OSTrafo})}}\)     \(\substack{\text{\normalsize\(L^2(\R_+,\C,\cT \nu)\)} \\ \text{\normalsize (\fref{cor:OSTrafoFromMeasures})}}\)     \(\substack{\text{\normalsize\(L^2(\R_+,\C,2\lambda_1)\)} \\ \text{\normalsize (\fref{cor:OSTrafoFromMeasures},} \\ \text{\normalsize \fref{ex:Lebesgue})}}\)

§.§ Extending the Longo–Witten Theorem beyond standard pairs

In [16], given a non-degenerate standard pair \((\sV,U)\) on a complex Hilbert space \(\cH\), the authors consider the semigroup

\[\cE(\sV,U) \coloneqq \{A \in \U(\cH): A\sV \subeq \sV, \,(\forall t \in \R)\, U_t A = A U_t\}.\]

The authors give the following theorem characterizing this semigroup as the semigroup of symmetric inner functions (cf. def:Inner):
Let \(\cH\) be a complex Hilbert space and \((\sV,U)\) be a non-degenerate standard pair on \(\cH\). Further let \(\cK\) be the multiplicity space of the real reflection positive ROPG \((\cH,\sV,U,J_\sV)\) and let \(\psi: \cH \to \cH_\cK\) be an isomorphism between \((\cH,\sV,U,J_\sV)\) and \((\cH_\cK,\sV_\cK,S,\theta_{i \cdot \sgn})\). Then

\[\cE(\sV,U) = \psi^{-1} \circ M(\Inn(\C_+,B(\cK_\C))^\sharp) \circ \psi.\]

The goal of this section is to provide a generalization of this theorem from non-degenerate standard pairs to regular pairs. So we make the following definition:

Let \(\cH\) be a complex Hilbert space and \((\sV,U)\) be a regular pair. We set

\[\gls*{EVU} \coloneqq \{A \in \U(\cH): A\sV \subeq \sV, \,(\forall t \in \R)\, U_t A = A U_t\}.\]

Let \(\cH\) be a complex Hilbert space and \((\sV,U)\) be a regular pair on \(\cH\). Further let \(\cK\) be the multiplicity space of the real ROPG \((\cH,\sV,U)\) and let \(\psi: \cH \to \cH_\cK\) be an isomorphism between \((\cH,\sV,U)\) and \((\cH_\cK,\sV_\cK,S)\). Then

\[\cE(\sV,U) = \psi^{-1} \circ M(\Inn(\C_+,B(\cK_\C))^\sharp) \circ \psi.\]

Further, in the case \(\dim \cK < \infty\), let \(\left(W_t\right)_{t \in \R_+} \subeq \cE(\sV,U)\) be a strongly continuous one-parameter semigroup. Then, setting

\[F \coloneqq \psi \circ \partial W \circ \psi^{-1} \in \mathrm{Pick}(\C_+,B(\cK_\C))_\R\]

(cf. thm:OPGOfInnerFunctions) and extending \(W\) to a unitary one-parameter group \(\tilde W: \R \to \U(\cH)\), we have that \((\sV,\tilde W)\) is a regular pair, if and only if one of the following equivalent conditions hold:

* \(\Spec(F(z)) \subeq \C_+\) for all \(z \in \C_+\).

* \(\ker(x\textbf{1}-M_{F_*}) \cap H^2(\C_+,\cH) = \{0\}\) for all \(x \in \R\).

* \(\R \setminus D(F,x)\) has non-zero Lebesgue measure for all \(x \in \R\).

* \(D(F,x)\) is a Lebesgue zero-set for all \(x \in \R\).

* \(D(F,M)\) is a Lebesgue zero-set for all Lebesgue zero-sets \(M \subeq \R\).

(cf. def:DFM for the definition of \(D(F,x)\) and \(D(F,M)\)). In this case, the multiplicity space \(\cM\) of \((\cH,\sV,\tilde W)\) is given by

\[\cM \cong \begin{cases} \R^{\deg F} & \text{if } F \in \mathrm{Rat}(\cK_\C) \\ \ell^2(\N,\R) & \text{if } F \notin \mathrm{Rat}(\cK_\C).\end{cases}\]

(cf. def:RationalPick).

We can without loss of generality assume that \((\cH,\sV,U) = (\cH_\cK,\sV_\cK,S)\). Then the commutation relation \(S_tA=AS_t\), \(t \in \R\), is equivalent to \(A \in M(L^\infty(\R,B(\cK_\C)))\) by prop:SCommutant. Then \(A\sV_\cK \subeq \sV_\cK\), by prop:H2InclusionFunctions, is equivalent to \(A \in M(H^\infty(\C_+,B(\cK_\C))^\sharp)\) and then finally \(A \in \U(\cH)\) is equivalent to \(A \in M(\Inn(\C_+,B(\cK_\C))^\sharp)\).

The statement about the regularity of the pair \((\sV,\tilde W)\) follows by thm:SpecNonDeg and thm:SpecNonDegZeroSet. In this case we get the formula for the multiplicity space \(\cM\) by thm:BigLWTheoremReal.

§ HOLOMORPHIC FUNCTIONS ON THE UPPER HALF PLANE

§.§ Hardy spaces

In this section we will introduce the concept of Hardy spaces and collect some basic facts about them. We will restrict ourselves mostly to Hardy spaces on the upper half plane \(\C_+\), but everything can analogously also be done for the lower half plane \(\C_-\). We start by the definition of a Hardy space:

For a function \(f \in \cO(\C_+)\), we set

\[\left\lVert f\right\rVert_{H^p} \coloneqq \sup_{y > 0} \left(\int_\R \left|f(x+iy)\right|^p \,dx\right)^{\frac 1p} \quad \text{for } 1 \leq p < \infty\]

\[\left\lVert f\right\rVert_{H^\infty} \coloneqq \sup_{z \in \C_+} \left|f(z)\right|.\]

Now, for \(1 \leq p \leq \infty\), we define the Hardy space \(\gls*{Hp}\) by

\[H^p(\C_+) \coloneqq \{f \in \cO(\C_+) : \left\lVert f\right\rVert_{H^p} < \infty\}.\]

(cf. <cit.>)
For \(1 \leq p \leq \infty\), the Hardy space \(H^p(\C_+)\) is a Banach space. Further \(H^2(\C_+)\) becomes a Hilbert space when endowed with the scalar product

\[\braket*{f}{g}_{H^2} \coloneqq \lim_{y \downarrow 0} \int_\R \overline{f(x+iy)} g(x+iy) dx, \quad f,g \in H^2(\C_+,\cH).\]

(cf. <cit.>)
For \(1 \leq p \leq \infty\), for every \(f \in H^p(\C_+)\), the non-tangential limit

\[f_*(x) \coloneqq \lim_{\epsilon \downarrow 0} f(x+i\epsilon)\]

exists for almost every \(x \in \R\) and defines a function \(f_* \in L^p(\R,\C)\) with \(\left\lVert f_*\right\rVert_p = \left\lVert f\right\rVert_{H^p}\). In particular, for the scalar product on \(H^2(\C_+)\), one has

\[\braket*{f}{g}_{H^2} = \braket*{f_*}{g_*} \quad \forall f,g \in H^2(\C_+).\]

This theorem shows that we have an isometric embedding

\begin{equation*}
\iota_p: H^p\left(\C_+\right) \to L^p\left(\R,\C\right), \quad f \mapsto f_*.
\end{equation*}

From now on, we will identify \(H^p\left(\C_+\right)\) with \(\iota_p\left(H^p\left(\C_+\right)\right) \subeq L^p\left(\R,\C\right)\) and we will write \(f(x)\) instead of \(f_*(x)\) for \(f \in H^p\left(\C_+\right)\) and \(x \in \R\).
(cf. <cit.>, <cit.>)
Let \(1 \leq p \leq \infty\). Then, for every \({f \in H^p(\C_+,\cH) \setminus \{0\}}\), one has

\[\int_{\R}\frac{|\log |f(x)||}{1+x^2} \,dx < \infty,\]

so especially

\[f(x) \neq 0\]

for almost every \(x \in \R\).

After considering Hardy spaces in general, we will now focuse on the Hardy space \(H^2(\C_+)\). \(H^2(\C_+)\) as a Hilbert space is an example of a so called reproducing kernel Hilbert space, defined as follows:

Let \(X\) be an arbitrary set and \(H\) be a Hilbert space that is a subset of the complex valued functions on \(X\). We say that \(H\) is a reproducing kernel Hilbert space if the linear functionals

\begin{equation*}
\ev_x: H \to \C, \quad f \mapsto f\left(x\right)
\end{equation*}

are continuous for every \(x \in X\). Then, by the Riesz Representation Theorem, for every \(x \in X\), there exists a function \(Q_x \in H\) such that

\begin{equation*}
f\left(x\right) = \ev_x\left(f\right) = \braket*{Q_x}{f}.
\end{equation*}

Let \(H\) be a reproducing kernel Hilbert space. Then

\begin{equation*}
\overline{\mathrm{span}\left\lbrace Q_x: x \in X\right\rbrace} = H.
\end{equation*}

Let \(f \in \left\lbrace Q_x: x \in X\right\rbrace^\perp\). Then for every \(x \in X\) we have

\begin{equation*}
0 = \braket*{Q_x}{f} = f\left(x\right)
\end{equation*}

and therefore \(f=0\).

(cf. <cit.>)
The Hilbert space \(H^2\left(\C_+\right)\) is a reproducing kernel Hilbert space and its reproducing kernel functions are given by the Szegö kernel

\begin{equation*}
\gls*{Qw}: \C_+ \to \C, \quad z \mapsto \frac{1}{2\pi} \cdot \frac{i}{z-\overline{w}}, \qquad w \in \C_+.
\end{equation*}

Further, for \(w \in \C_+\), we have

\begin{equation*}
\left\lVert Q_w\right\rVert_2 = \frac 1{2\sqrt{\pi \mathrm{Im}(w)}}.
\end{equation*}

Obviously, \(Q_w\) is holomorphic and we have

\begin{align*}
\left\lVert Q_w\right\rVert_2^2 &= \sup_{y>0} \int_\R \left|Q_w(x+iy)\right|^2 dx = \sup_{y>0} \frac 1{(2\pi)^2}\int_\R \frac 1{(x - \mathrm{Re}(w))^2+(y+\mathrm{Im}(w))^2} dx
\\&= \sup_{y>0} \frac 1{(2\pi)^2} \frac{\pi}{y+\mathrm{Im}(w)} = \frac 1{(2\pi)^2} \frac{\pi}{\mathrm{Im}(w)} = \frac{1}{4 \pi \mathrm{Im}(w)} < \infty,
\end{align*}

so \(Q_w \in H^2\left(\C_+\right)\) with

\[\left\lVert Q_w\right\rVert_2 = \frac 1{2\sqrt{\pi \mathrm{Im}(w)}}.\]

Further, by Cauchy's Integral Formula, we have

\begin{equation*}
\braket*{Q_w}{f} = \frac{1}{2 \pi i} \int_{\R} \frac{1}{p-w}f\left(p\right) dp = f\left(w\right),
\end{equation*}

which shows, that \(Q_w\) is the reproducing kernel function.

Let \(f \in H^2\left(\C_+\right)\) and \(\left(f_n\right)_{n \in \N} \subseteq H^2\left(\C_+\right)\) such that \(\left\lVert f_n-f\right\rVert_2 \xrightarrow{n \to \infty} 0\). Then, for every \(\beta>0\), the sequence \(\left(f_n\right)_{n \in \N}\) converges uniformly to \(f\) on the set

\[I_\beta \coloneqq \left\lbrace z \in \C: \mathrm{Im}\left(z\right) \geq \beta\right\rbrace.\]

By prop:H2RKHS, we have

\begin{align*}
\sup_{z \in I_\beta}\left|f_n\left(z\right)-f\left(z\right)\right| &= \sup_{z \in I_\beta}\left|\braket*{Q_z}{f_n-f}\right| \leq \sup_{z \in I_\beta}\left\lVert Q_z\right\rVert_2 \cdot \left\lVert f_n-f\right\rVert_2
\\&= \sup_{z \in I_\beta}\frac{1}{2\sqrt{\pi \mathrm{Im}(z)}} \cdot \left\lVert f_n-f\right\rVert_2 = \frac{1}{2\sqrt{\pi \beta}} \cdot \left\lVert f_n-f\right\rVert_2 \xrightarrow{n \to \infty} 0
\end{align*}

and therefore the convergence is uniformly on \(I_\beta\).

After considering Hardy spaces of scalar valued functions, we now want to introduce Hardy spaces with values in a Hilbert space or in the bounded operators of some Hilbert space respectively:

Let \(\cH\) be a complex Hilbert space.

* For an open subset \(U \subset \C\) we write \(\cO(U,\cH)\) for the set of holomorphic functions on \(U\) with values in \(\cH\) and \(\cO(U,B(\cH))\) for the set of holomorphic functions on \(U\) with values in \(B(\cH)\).

* For a function \(f \in \cO(\C_+,\cH)\), we set

\[\left\lVert f\right\rVert_{H^2} \coloneqq \sup_{y > 0} \left(\int_\R \left\lVert f(x+iy)\right\rVert^2 \,dx\right)^{\frac 12}\]

and define

\[\gls*{H2H} \coloneqq \{f \in \cO(\C_+,\cH) : \left\lVert f\right\rVert_{H^2} < \infty\}.\]

* For a function \(f \in \cO(\C_+,B(\cH))\), we set

\[\left\lVert f\right\rVert_{H^\infty} \coloneqq \sup_{z \in \C_+} \left\lVert f(z)\right\rVert\]

and define

\[\gls*{H8BH} \coloneqq \{f \in \cO(\C_+,B(\cH)) : \left\lVert f\right\rVert_{H^\infty} < \infty\}.\]

As before in the scalar valued case \(H^\infty(\C_+,B(\cH))\) is a Banach space and \(H^2(\C_+,\cH)\) becomes a Hilbert space when endowed with the scalar product

\[\braket*{f}{g}_{H^2} \coloneqq \lim_{y \downarrow 0} \int_\R \braket*{f(x+iy)}{g(x+iy)}_\cH \,dx, \quad f,g \in H^2(\C_+,\cH)\]

denoting by \(\braket*{\cdot}{\cdot}_\cH\) the scalar product on \(\cH\).

By thm:HpBoundaryValues, for every \(f \in H^2(\C_+,\cH)\) and \(v \in \cH\), the limit

\[f_v(x) \coloneqq \lim_{\epsilon \downarrow 0} \braket*{v}{f(x+i\epsilon)}_\cH\]

exists for almost every \(x \in \R\). Analogously, for every \(f \in H^\infty(\C_+,B(\cH))\) and every trace-class operator \(T \in B_1(\cH)\), the limit

\[f_T(x) \coloneqq \lim_{\epsilon \downarrow 0} \mathrm{Tr}(T f(x+i\epsilon))\]

exists for almost every \(x \in \R\). This shows that, by taking weak limits, we can identify \(H^2(\C_+,\cH)\) with a subspace of \(L^2(\R,\cH)\) and \(H^\infty(\C_+,B(\cH))\) with a subspace of \(L^\infty(\R,B(\cH))\) respectively.

Let \(f \in H^\infty(\C_+)\) with \(f(x) \in \R\) for almost every \(x \in \R\). Then \(f\) is constant.

Since \(f \in H^\infty(\C_+)\) and \(\exp: \C \to \C\) maps bounded sets to bounded sets, we have

\[g_+ \coloneqq e^{if} \in H^\infty(\C_+) \quad \text{and} \quad g_- \coloneqq e^{-if} \in H^\infty(\C_+)\]

\[|g_+(x)| = |g_-(x)| = 1\]

for almost every \(x \in \R\). This implies that

\[|g_+(z)| \leq \left\lVert g_+\right\rVert_\infty = 1 \quad \text{and} \quad |g_-(z)| \leq \left\lVert g_-\right\rVert_\infty = 1 \qquad \forall z \in \C_+.\]

This yields

\[\Im(f(z)) \geq 0 \quad \text{and} \quad \Im(f(z)) \leq 0 \qquad \forall z \in \C_+\]

and therefore \(f(z) \in \R\) for every \(z \in \R\). This, by the Cauchy–Riemann equations, implies that \(f\) is constant.

Let \(\cH\) be a complex Hilbert space. Then

\[H^\infty(\C_+,B(\cH)) \cap H^\infty(\C_-,B(\cH)) = B(\cH) \cdot \textbf{1}.\]

We first consider the scalar valued case, so let \(f \in H^\infty(\C_+) \cap H^\infty(\C_-)\). Then \(f \in H^\infty(\C_+)\) and \(f \in H^\infty(\C_-)\), where the latter implies that \(\overline{f} \in H^\infty(\C_+)\). This yields

\[\Re(f),\Im(f) \in H^\infty(\C_+),\]

which, by prop:HInftyRealBoundary implies that \(\Re(f)\) and \(\Im(f)\) are constant and therefore also \(f\) is constant, so \(f = c \cdot \textbf{1}\) for some \(c \in \C\).

We now consider the operator-valued case, so let \(f \in H^\infty(\C_+,B(\cH)) \cap H^\infty(\C_-,B(\cH))\). Then, for every \(v,w \in \cH\), we have

\[\braket*{v}{f(\cdot)w} \in H^\infty(\C_+) \cap H^\infty(\C_-) = \C \cdot \textbf{1}.\]

This implies that \(f \in B(\cH) \cdot \textbf{1}\).

Let \(\cH\) be a complex Hilbert space and let \(f \in L^\infty(\R,B(\cH))\). Then

\[M_fH^2(\C_+,\cH) \subeq H^2(\C_+,\cH),\]

if and only if \(f \in H^\infty(\C_+,B(\cH))\).

Let \(f \in H^\infty(\C_+,B(\cH))\) and \(g \in H^2(\C_+,\cH)\). Then obviously \(M_f g = f \cdot g \in \cO(\C_+,\cH)\). Further

\begin{align*}
\left\lVert M_f g\right\rVert_2^2 &= \left\lVert f \cdot g\right\rVert_2^2 = \sup_{y > 0} \int_\R \left\lVert f(x+iy) g(x+iy)\right\rVert^2 \,dx \leq \sup_{y > 0} \int_\R \left\lVert f(x+iy)\right\rVert^2 \cdot \left\lVert g(x+iy)\right\rVert^2 \,dx
\\&\leq \sup_{y > 0} \int_\R \left\lVert f\right\rVert_\infty^2 \cdot \left\lVert g(x+iy)\right\rVert^2 \,dx = \left\lVert f\right\rVert_\infty^2 \cdot \left\lVert g\right\rVert_2^2,
\end{align*}

so \(M_f g \in H^2(\C_+,\cH)\). This implies that

\[M_fH^2(\C_+,\cH) \subeq H^2(\C_+,\cH).\]

Now, for the converse, we first consider the scalar valued case. So let \(f \in L^\infty(\R,\C)\) with

\[M_fH^2(\C_+) \subeq H^2(\C_+).\]

This in particular implies

\[f \cdot Q_i \in H^2(\C_+).\]

Therefore, by <cit.>, we have

\[0 = \frac 1{2\pi i} \int_\R \frac{f(x) \cdot Q_i(x)}{x-\overline{z}} \,dx = \frac 1{4\pi^2} \int_\R \frac{f(x)}{(x-\overline{z})(x+i)} \,dx \quad \forall z \in \C_+,\]

which, by <cit.> implies that \(f \in H^\infty(\C_+)\).

We now consider the operator-valued case, so let \(f \in L^\infty(\R,B(\cH))\) with

\[M_fH^2(\C_+,\cH) \subeq H^2(\C_+,\cH).\]

Then, for every \(v,w \in \cH\), defining

\[f_{v,w} \coloneqq \braket*{v}{f(\cdot)w} \in L^\infty(\R,\C),\]

we get

\[M_{f_{v,w}}H^2(\C_+) = \braket*{v}{M_f (H^2(\C_+) \otimes w)} \subeq \braket*{v}{M_fH^2(\C_+,\cH)} \subeq \braket*{v}{H^2(\C_+,\cH)} = H^2(\C_+),\]

\[f_{v,w} \in H^\infty(\C_+).\]

This means that, for \(z \in \C_+\), we can define the map

\[F_z: \cH \times \cH \to \C, \quad (v,w) \mapsto f_{v,w}(z),\]

which is sesquilinear by construction. Further, for every \(z \in \C_+\), we have

\[|F_z(v,w)| = |f_{v,w}(z)| \leq \left\lVert f_{v,w}\right\rVert_\infty = \left\lVert \braket*{v}{f(\cdot)w}\right\rVert_\infty \leq \left\lVert f\right\rVert_\infty \cdot \left\lVert v\right\rVert \cdot \left\lVert w\right\rVert, \quad \forall v,w \in \cH,\]

so there exists \(A(z) \in B(\cH)\) with \(\left\lVert A(z) \right\rVert \leq \left\lVert f\right\rVert_\infty\) such that

\[F_z(v,w) = \braket*{v}{A(z) w}, \quad \forall v,w \in \cH.\]

\[A: \C_+ \to B(\cH), \quad z \mapsto A(z)\]

is a bounded function, for which all the functions

\[\braket*{v}{A(\cdot)w} = f_{v,w}, \quad v,w \in \cH\]

are holomorphic. This, by <cit.>, implies that \(A\) is holomorphic and therefore \(A \in H^\infty(\C_+,B(\cH))\). Now, by construction, for every \(v,w \in \cH\) and almost every \(x \in \R\), we have

\[\braket*{v}{A(x)w} = f_{v,w}(x) = \braket*{v}{f(x)w},\]

which implies that \(f = A \in H^\infty(\C_+,B(\cH))\).

Let \(\cH\) be a complex Hilbert space and let \(f,g \in L^\infty(\R,B(\cH))\) such that \(f(x),g(x) \in U(\cH)\) for almost all \(x \in \R\). Then, one has

\[M_f H^2(\C_+,\cH) = M_g H^2(\C_+,\cH),\]

if and only if there exists \(u \in \U(\cH)\) such that

\[f = g \cdot u.\]

By multiplying with \(M_{g^{-1}}\) and \(M_{f^{-1}}\) respectively from the left, we get

\[M_{g^{-1}f} H^2(\C_+,\cH) = H^2(\C_+,\cH) \quad \text{and} \quad M_{f^{-1}g} H^2(\C_+,\cH) = H^2(\C_+,\cH).\]

By prop:H2InclusionFunctions this implies \(g^{-1}f,f^{-1}g \in H^\infty(\C_+,B(\cH))\). On the other hand

\[g^{-1}f = (f^{-1}g)^{-1} = (f^{-1}g)^* \in H^\infty(\C_-,B(\cH)),\]

so, by cor:HInftyIntersection, we get

\[g^{-1}f \in H^\infty(\C_+,B(\cH)) \cap H^\infty(\C_-,B(\cH)) = B(\cH) \cdot \textbf{1}\]

and therefore, there exists \(u \in B(\cH)\) such that

\[f = g \cdot u.\]

That \(u \in \U(\cH)\) follows from the fact that \(f\) and \(g\) have unitary values almost everywhere.

We finish this section by some density theorems:

For \(t \in \R\) we consider the function \(e_t \in L^\infty(\R,\C)\) with

\[e_t(x) \coloneqq e^{itx}, \quad x \in \R.\]

Then one has

\[\overline{\spann \{e_t: t \in \R\}} = L^\infty(\R,\C) \qquad \text{and} \qquad \overline{\spann \{e_t: t \in \R_+\}} = H^\infty(\C_+),\]

where the closures are taken with respect to the weak-\(*\)-topology on \(L^\infty(\R,\C)\).

Let \(f \in L^1(\R,\C)\) with

\[0 = \int_\R e_t(x) \cdot f(x) \,dx = \hat f(-t) \quad \forall t \in \R.\]

Then \(\hat f = 0\) and therefore \(f = 0\), since the Fourier transform is injective. This shows that

\[\overline{\spann \{e_t: t \in \R\}} = L^\infty(\R,\C).\]

The second statement follows by <cit.>.

We need a similiar statement to the one of prop:etDense also for the Hilbert space \(L^2(\T,\C)\) and it's corresponding Hardy space on the unit disc \(\D\). For this we first have to introduce Hardy spaces on the disc:

* For a function \(f \in \cO(\D)\), we set

\[\left\lVert f\right\rVert_{H^p} \coloneqq \sup_{0 < r < 1} \left(\int_\T \left|f(rz)\right|^p \,dz\right)^{\frac 1p} \quad \text{for } 1 \leq p < \infty\]

\[\left\lVert f\right\rVert_{H^\infty} \coloneqq \sup_{z \in \D} \left|f(z)\right|.\]

Now, for \(1 \leq p \leq \infty\), we define the Hardy space \(H^p(\D)\) by

\[\gls*{HpD} \coloneqq \{f \in \cO(\D) : \left\lVert f\right\rVert_{H^p} < \infty\}.\]

* For a function \(f \in \cO(\C_+,\cH)\), we set

\[\left\lVert f\right\rVert_{H^2} \coloneqq \sup_{0 < r < 1} \left(\int_\T \left\lVert f(rz)\right\rVert^2 \,dz\right)^{\frac 12}\]

and define

\[H^2(\D,\cH) \coloneqq \{f \in \cO(\D,\cH) : \left\lVert f\right\rVert_{H^2} < \infty\}.\]

* For a function \(f \in \cO(\D,B(\cH))\), we set

\[\left\lVert f\right\rVert_{H^\infty} \coloneqq \sup_{z \in \D} \left\lVert f(z)\right\rVert\]

and define

\[H^\infty(\D,B(\cH)) \coloneqq \{f \in \cO(\D,B(\cH)) : \left\lVert f\right\rVert_{H^\infty} < \infty\}.\]

Similar to the Hardy spaces on the upper half plane \(\C_+\), by taking boundary values

\[f(z) \coloneqq \lim_{r \uparrow 1} f(rz), \quad z \in \T,\]

one gets an isometric embedding of \(H^p(\D)\) into the Banach space \({L^p(\T,\C) \coloneqq L^p(\T,\C,\mu)}\) (cf. <cit.>), where by \(\mu\) we denote the unique Haar measure on the group \(\T\) normalized by \(\mu(\T) = 1\). In the same fashion one also has embeddings of \(H^2(\D,\cH)\) into \(L^2(\T,\cH)\) and of \(H^\infty(\D,B(\cH))\) into \(L^\infty(\T,B(\cH))\).

One has

\[\overline{\spann \{\Id_\T^n: n \in \Z\}} = L^\infty(\T,\C) \qquad \text{and} \qquad \overline{\spann \{\Id_\T^n: n \in \N_0\}} = H^\infty(\D),\]

where the closure is taken with respect to the weak-\(*\)-topology on \(L^\infty(\T,\C)\).

Let \(f \in L^1(\T,\C)\) with

\[0 = \int_\R x^n \cdot f(x) \,dx = \hat f(-n) \quad \forall n \in \Z.\]

Then \(\hat f = 0\) and therefore \(f = 0\), since the Fourier transform is injective. This shows that

\[\overline{\spann \{\Id_\T^n: n \in \Z\}} = L^\infty(\T,\C).\]

The second statement follows by <cit.>.

§.§ Outer functions

In this section we introduce the concept of outer functions and present some basic facts about them, as well as about their interplay with Hardy spaces. We start with the definition of outer functions:
(cf. <cit.>)

* For \(C \in \T\) and an almost everywhere defined function \(K: \R \to \R_{\geq 0}\) with

\begin{equation*}
\gls*{IK} \coloneqq \int_\R \frac{\left|\log\left(K\left(p\right)\right)\right|}{1 + p^2} \,dp < \infty
\end{equation*}

we define the function \(\Out(C,K) \in \cO(\C_+)\) by

\begin{equation*}
\Out(C,K)\left(z\right) = C \cdot \exp\left(\frac{1}{\pi i} \int_\R \left[\frac{1}{p-z} - \frac{p}{1 + p^2}\right] \log\left(K\left(p\right)\right) dp\right), \quad z \in \C_+
\end{equation*}

and call such a function an outer function.

* We set \(\gls*{OutK} \coloneqq \Out(1,K)\).

* We write \(\gls*{Out}\) for the set of outer functions on \(\C_+\) and set

\[\Out^2(\C_+) = \Out(\C_+) \cap H^2(\C_+).\]

(cf. <cit.>, <cit.>)
Let \(C \in \T\) and \(K: \R \to \R_+\) with \(I(K) < \infty\). Then the limit

\[\Out(C,K)(x) \coloneqq \lim_{\epsilon \downarrow 0} \Out(C,K)(x+i\epsilon)\]

exists for almost all \(x \in \R\) and satisfies

\[|\Out(C,K)(x)| = K(x).\]

Further, one has \(\Out(C,K) \in H^p(\C_+)\), if and only if \(K \in L^p(\R,\C)\).

Let \(F \in H^2(\C_+)\) and consider, for \(t \in \R\), the unitary operator \(S_t \coloneqq M_{e_t}\) with \(e_t(x) \coloneqq e^{itx}\), \(x \in \R\).
Then \(F \in \Out^2(\C_+)\), if and only if

\[H^2(\C_+) = \oline{\Spann S(\R_+) F}.\]

For every outer function \(F \in \Out^2(\C_+)\) one has

\[\overline{\spann \,S(\R) F} = \overline{\left(H^\infty(\C_+) + H^\infty(\C_-)\right)F} = \overline{L^\infty(\R,\C)F} = L^2(\R,\C).\]

By thm:outerSpan, we have

\[\overline{\spann \,S(\R) F} \supeq \overline{\bigcup_{t \in \R} \overline{\spann \,S(\R_{> t}) F}} = \overline{\bigcup_{t \in \R} S_t \overline{\spann \,S(\R_+) F}} = \overline{\bigcup_{t \in \R} S_t H^2(\C_+)} = L^2(\R,\C),\]

using thm:LaxPhillipsComplex in the last step.

\[\spann \,\{e_t: t \in \R\} = \spann \,\{e_t: t \in \R_+\} + \spann \,\{e_t: t \in \R_{\leq 0}\} \subeq H^\infty(\C_+) + H^\infty(\C_-) \subeq L^\infty(\R,\C),\]

we get

\[L^2(\R,\C) \subeq \overline{\spann \,S(\R) F} \subeq  \overline{\left(H^\infty(\C_+) + H^\infty(\C_-)\right)F} \subeq \overline{L^\infty(\R,\C)F} \subeq L^2(\R,\C),\]

which implies the statement.

We now show some basic properties of outer functions:

Let \(c \in \R_+\) and let \(f,g: \R \to \R_{\geq 0}\) with \(I(f) < \infty\) and \(I(g) < \infty\). Then

\[I(c \cdot f) < \infty, \quad I(f \cdot g) < \infty \quad \text{and} \quad I\left(\frac{f}{g}\right) < \infty\]

\[\Out(c \cdot f) = c \cdot \Out(f), \quad \Out(f \cdot g) =  \Out(f) \cdot \Out(g) \quad \text{and} \quad \Out\left(\frac{f}{g}\right) = \frac{\Out(f)}{\Out(g)}.\]

For \(p \in \R\) one has

\[\left|\log\left((c \cdot f)\left(p\right)\right)\right| = \left|\log\left(c\right) + \log\left(f\left(p\right)\right)\right| \leq \left|\log\left(c\right)\right| + \left|\log\left(f\left(p\right)\right)\right|\]

\[\left|\log\left((f \cdot g)\left(p\right)\right)\right| = \left|\log\left(f\left(p\right)\right) + \log\left(g\left(p\right)\right)\right| \leq \left|\log\left(f\left(p\right)\right)\right| + \left|\log\left(g\left(p\right)\right)\right|\]

\[\left|\log\left(\left(\frac fg\right)\left(p\right)\right)\right| = \left|\log\left(f\left(p\right)\right) - \log\left(g\left(p\right)\right)\right| \leq \left|\log\left(f\left(p\right)\right)\right| + \left|\log\left(g\left(p\right)\right)\right|,\]

which immediately implies

\[I(c \cdot f) < \infty, \quad I(f \cdot g) < \infty \quad \text{and} \quad I\left(\frac{f}{g}\right) < \infty.\]

The second statement then follows immediately from the definition of outer functions.

Let \(K: \R \to \R_{\geq 0}\) be a symmetric function with \(I(K) < \infty\).

\[\overline{\Out(K)(-\overline{z})} = \Out(K)(z) \quad \forall z \in \C_+.\]

\[\Out(K)(i\lambda) = \exp\left(\frac{1}{\pi} \int_\R \frac{\lambda}{p^2+\lambda^2} \log\left(K\left(p\right)\right) dp\right) \in \R_+ \quad \forall \lambda \in \R_+.\]

For every \(z \in \C_+\), we have

\begin{align*}
\overline{\Out(K)(-\overline{z})} &= \overline{\exp\left(\frac{1}{\pi i} \int_\R \left[\frac{1}{p+\overline{z}} - \frac{p}{1 + p^2}\right] \log\left(K\left(p\right)\right) dp\right)}
\\&= \exp\left(\frac{-1}{\pi i} \int_\R \left[\frac{1}{p+z} - \frac{p}{1 + p^2}\right] \log\left(K\left(p\right)\right) dp\right)
\\&= \exp\left(\frac{-1}{\pi i} \int_\R \left[\frac{1}{-p+z} - \frac{-p}{1 + (-p)^2}\right] \log\left(K\left(-p\right)\right) dp\right)
\\&=\exp\left(\frac{1}{\pi i} \int_\R \left[\frac{1}{p-z} - \frac{p}{1 + p^2}\right] \log\left(K\left(p\right)\right) dp\right) = F_K(z).
\end{align*}

Further, since \(K\) is symmetric, also \(\sqrt{K}\) is symmetric and therefore also

\[\overline{\Out\big(\sqrt{K}\big)(-\overline{z})} = \Out\big(\sqrt{K}\big)(z) \quad \forall z \in \C_+.\]

This in particular yields

\[\Out\big(\sqrt{K}\big)(i\lambda) = \overline{\Out\big(\sqrt{K}\big)(-\overline{i\lambda})} = \overline{\Out\big(\sqrt{K}\big)(i\lambda)} \quad \forall \lambda \in \R_+.\]

Then, by lem:OuterHomo, for \(\lambda \in \R_+\), we have

\begin{align*}
\Out(K)(i\lambda) &= \Out\big(\sqrt{K}\big)(i\lambda) \cdot \Out\big(\sqrt{K}\big)(i\lambda)
\\&=  \Out\big(\sqrt{K}\big)(i\lambda) \cdot \overline{\Out\big(\sqrt{K}\big)(i\lambda)} = \left|\Out\big(\sqrt{K}\big)(i\lambda)\right|^2 \geq 0,
\end{align*}

so \(\Out(K)(i\lambda) \in \R_{\geq 0}\). Further, for every \(\lambda \in \R_+\), we have \(\Out(K)(i\lambda) \in \exp(\C) = \C^\times\), so

\[\Out(K)(i\lambda) \in \R_+.\]

Therefore, for \(\lambda \in \R_+\), we get

\begin{align*}
\Out(K)(i\lambda) &= \left|\Out(K)(i\lambda)\right| = \left|\exp\left(\frac{1}{\pi i} \int_\R \left[\frac{1}{p-i\lambda} - \frac{p}{1 + p^2}\right] \log\left(K\left(p\right)\right) dp\right)\right|
\\&= \exp\left(\mathrm{Re}\left(\frac{1}{\pi i} \int_\R \left[\frac{1}{p-i\lambda} - \frac{p}{1 + p^2}\right] \log\left(K\left(p\right)\right) dp\right)\right)
\\&= \exp\left(\mathrm{Re}\left(\frac{1}{\pi i} \int_\R \left[\frac{p+i\lambda}{p^2+\lambda^2} - \frac{p}{1 + p^2}\right] \log\left(K\left(p\right)\right) dp\right)\right)
\\&= \exp\left(\frac{1}{\pi} \int_\R \frac{\lambda}{p^2+\lambda^2} \log\left(K\left(p\right)\right) dp\right). \qedhere
\end{align*}

Finally, we provide some concrete examples of outer functions:

Let \(\sqrt{\cdot}: \C_+ \to \C\) denote the square root on \(\C_+\) with \(\sqrt{\C_+} \subeq \C_+\). Then, for every \(z \in \C_+\), given the function \(K: \R \to \R_{\geq 0}\) with

* \(K(p) = |p|\), one has \(\Out(K)(z) = -iz\).

* \(K(p) = \frac 1{|p|}\), one has \(\Out(K)(z) = \frac i{z}\).

* \(K(p) = \frac 1{\sqrt{|p|}}\), one has \(\Out(K)(z) = \frac {i+1}{\sqrt{2z}}\).

* \(K(p) = 1+p^2\), one has \(\Out(K)(z) = -(z+i)^2\).

* \(K(p) = \frac {\left|p\right|}{1+p^2}\), one has \(\Out(K)(z) = \frac {iz}{(z+i)^2}\).

* \(K(p) = \frac 1{\sqrt{1+p^2}}\), one has \(\Out(K)(z) = \frac i{z+i}\).

* We have

\begin{align*}
\int_{\R_+} \frac{\log\left(p\right)}{p^2+1} \,dp = \int_\R \frac{\log\left(e^x\right)}{e^{2x}+1} \cdot e^x \,dx = \int_\R \frac{x}{e^x+e^{-x}} \,dx = 0,
\end{align*}

where the last equality follows from the fact that the integrand is skew-symmetric.

Then, for \(\lambda \in \R_+\), by lem:OuterSymmetric, we have

\begin{align*}
\Out(K)(i\lambda) &= \exp\left(\frac{1}{\pi} \int_\R \frac{\lambda}{p^2+\lambda^2} \log\left|p\right| dp\right) = \exp\left(\frac{2}{\pi} \int_{\R_+} \frac{\lambda}{p^2+\lambda^2} \log\left(p\right) dp\right)
\\&= \exp\left(\frac{2}{\pi} \int_{\R_+} \frac{\lambda^2}{(\lambda p)^2+\lambda^2} \log\left(\lambda p\right) dp\right) = \exp\left(\frac{2}{\pi} \int_{\R_+} \frac{1}{p^2+1} \log\left(\lambda p\right) dp\right)
\\&= \exp\left(\frac{2}{\pi} \int_{\R_+} \frac{1}{p^2+1} \log\left(\lambda\right) dp + \frac{2}{\pi} \int_{\R_+} \frac{1}{p^2+1} \log\left(p\right) dp\right)
\\&= \exp\left(\log\left(\lambda\right) + 0\right) = \lambda = -i(i\lambda).
\end{align*}

This shows that the holomorphic functions \(\Out(K)\) and \(z \mapsto -iz\) coincide on \(i\R_+\) and are therefore equal.

* By lem:OuterHomo and (a) we have

\[\Out(K)(z) = \frac 1{-iz} = \frac iz, \quad z \in \C_+.\]

* By lem:OuterHomo and (b), for \(z \in \C_+\), we have

\[\Out(K)(z) \cdot \Out(K)(z) = \frac iz,\]

\[\Out(K)(z) = \pm \frac{\frac {i+1}{\sqrt{2}}}{\sqrt{z}} = \pm \frac {i+1}{\sqrt{2z}}.\]

The statement then follows by lem:OuterSymmetric since

\[+ \frac {i+1}{\sqrt{2i}} = 1 \in \R_+.\]

* We define the function

\[g: (-1,\infty) \to \R, \quad \lambda \mapsto \int_{\R_+} \frac{1}{p^2+1} \left[\log\left(1+\lambda^2p^2\right) - 2\log\left(\lambda+1\right)\right] dp.\]

For \(\lambda \neq 1\), we have

\begin{align*}