Regular one-parameter groups,
reflection positivity and their application to

Hankel operators and standard subspaces

Reguläre Einparameter-Gruppen,
Reflexionspositivität und deren Anwendung auf

Hankel-Operatoren und Standard-Unterräume

Der Naturwissenschaftlichen Fakultät

der Friedrich-Alexander-Universität

Erlangung des Doktorgrades Dr. rer. nat.

vorgelegt von

Jonas Schober

Als Dissertation genehmigt von der Naturwissenschaftlichen Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg.

Tag der mündlichen Prüfung: 2. August 2023

Gutachter: Prof. Dr. Karl-Hermann Neeb

Prof. Dr. Gandalf Lechner

Prof. Dr. Yoh Tanimoto

§ ZUSAMMENFASSUNG

Standard-Unterräume sind viel studierte Objekte in der algebraischen Quantenfeldtheorie (AQFT). Gegeben einen Standard-Unterraum \(\sV\) in einem Hilbertraum \(\cH\), ist man interessiert an unitären Einparameter-Gruppen auf \(\cH\) für welche \(U_t \sV \subeq \sV\) für alle \(t \in \R_+\) gilt. Falls \((\sV,U)\) ein nicht-degeneriertes Standard-Paar auf \(\cH\) ist, d.h. der selbstadjungierte infinitesimale Erzeuger von \(U\) ist ein positiver Operator mit trivialem Kern, haben wir zwei klassische Resultate mit dem Satz von Borchers, welcher nicht-degenerierte Standard-Paare mit Darstellungen der affinen Gruppe \(\mathrm{Aff}(\R)\) mit positiver Energie in Verbindung setzt und dem Satz von Longo–Witten, laut welchem die Halbgruppe der unitären Endomorphismen von \(\sV\) mit der Halbgruppe der symmetrischen operatorwertigen inneren Funktionen auf der oberen Halbebene identifiziert werden kann.

In dieser Arbeit beweisen wir Sätze ähnlich denen von Borchers und Longo–Witten auch für allgemeinere Fälle von unitären Einparameter-Gruppen ohne die Annahme, dass deren infinitesimaler Erzeuger positiv ist. Wir wollen diese Annahme durch die schwächere Annahme ersetzen, dass das Tripel \((\cH,\sV,U)\) eine sogenannte reelle reguläre Einparameter-Gruppe ist.

Nach Bereitstellung der grundlegenden Theorie von regulären Einparameter-Gruppen, werden wir reguläre Einparameter-Gruppen untersuchen, die infinitesimal erzeugt werden von operatorwertigen Pick-Funktionen. Wir werden Kriterien bereitstellen, um zu prüfen, wann eine Pick-Funktion eine reguläre Einparameter-Gruppe generiert und in diesem Fall werden wir Formeln für den Multiplizitätenraum der regulären Einparameter-Gruppe bereitstellen und zeigen, dass die Dimension des Multiplizitätenraumes mit der Komposition von Pick-Funktionen verträglich ist.

Wir werden auch reflexionspositive reguläre Einparameter-Gruppen untersuchen und eine Normalform für diese angeben. Wir tun dies, indem wir sie in Verbindung setzen mit reflexionspositiven Hilberträumen der Form \((L^2(\R,\cK),H^2(\C_+,\cK),\theta_h)\) mit einem komplexen Hilbertraum \(\cK\) und einer Funktion \(h \in L^\infty(\R,\U(\cK))\), wobei die Involution \(\theta_h\) auf \(L^2(\R,\cK)\) durch \({(\theta_h f)(x) = h(x) \cdot f(-x)}\), \(x \in \R\), gegeben ist. In dem multiplizitätsfreien Fall \(\cK = \R\) geben wir eine vollständige Klassifikation aller Funktionen \({h \in L^\infty(\R,\T)}\), für welche das Tripel \((L^2(\R,\C),H^2(\C_+),\theta_h)\) ein maximaler reflexionspositiver Hilbertraum ist. Außerdem geben wir eine explizite Beschreibung der Osterwalder–Schrader–Transformation dieser reflexionspositiven Hilberträume.

Wir werden diese Resultate anwenden, um zu beweisen, dass jeder positive kontraktive Hankel-Operator auf \(H^2(\C_+)\) zu einer Involution \(\theta_h\) auf \(L^2(\R,\C)\) erweitert werden kann, für welche das Tripel \((L^2(\R,\C),H^2(\C_+),\theta_h)\) ein maximaler reflexionspositiver Hilbertraum ist.

Abschließend wenden wir unsere Resultate über (reflexionspositive) reguläre Einparameter-Gruppen auf den Kontext von Standard-Unterräumen an und benutzen sie, um Analoga für den Satz von Borchers und den Satz von Longo–Witten auch für solche Paare \((\sV,U)\) zu beweisen, für welche das Tripel \((\cH,\sV,U)\) eine reelle reguläre Einparameter-Gruppe ist bzw. für welche das Quadrupel \((\cH,\sV,U,J_\sV)\) eine reelle reflexionspositive reguläre Einparameter-Gruppe ist.

§ ABSTRACT

Standard subspaces are a well studied object in algebraic quantum field theory (AQFT). Given a standard subspace \(\sV\) of a Hilbert space \(\cH\), one is interested in unitary one-parameter groups on \(\cH\) with \(U_t \sV \subeq \sV\) for every \(t \in \R_+\). If \((\sV,U)\) is a non-degenerate standard pair on \(\cH\), i.e. the self-adjoint infinitesimal generator of \(U\) is a positive operator with trivial kernel, two classical results are given by Borchers' Theorem, relating non-degenerate standard pairs to positive energy representations of the affine group \(\mathrm{Aff}(\R)\) and the Longo–Witten Theorem, stating the the semigroup of unitary endomorphisms of \(\sV\) can be identified with the semigroup of symmetric operator-valued inner functions on the upper half plane.

In this thesis we prove results similar to the theorems of Borchers and of Longo–Witten for a more general framework of unitary one-parameter groups without the assumption that their infinitesimal generator is positive. We replace this assumption by the weaker assumption that the triple \((\cH,\sV,U)\) is a so called real regular one-parameter group.

After providing some basic theory for regular one-parameter groups, we investigate regular one-parameter groups that are infinitesimally generated by operator-valued Pick functions. We provide criteria for a Pick function to generate a regular one-parameter group and in this case we provide a formula for the multiplicity space of the regular one-parameter group and show that the dimension of the multiplicity space is compatible with composition of Pick functions.

We also investigate reflection positive regular one-parameter groups and provide a normal form for them. We do this by linking them to reflection positive Hilbert spaces of the form \((L^2(\R,\cK),H^2(\C_+,\cK),\theta_h)\) with some complex Hilbert space \(\cK\) and some function \(h \in L^\infty(\R,\U(\cK))\), where the involution \(\theta_h\) on \(L^2(\R,\cK)\) is given by \({(\theta_h f)(x) = h(x) \cdot f(-x)}\), \(x \in \R\). In the multiplicity free case \(\cK = \R\) we give a full classification of all functions \({h \in L^\infty(\R,\T)}\) for which the triple \((L^2(\R,\C),H^2(\C_+),\theta_h)\) is a maximal reflection positive Hilbert space. Also we give an explicit description of the Osterwalder–Schrader transform of these reflection positive Hilbert spaces.

We apply these results to prove that every positive contractive Hankel operator on \(H^2(\C_+)\) can be extended to an involution \(\theta_h\) on \(L^2(\R,\C)\) for which the triple \((L^2(\R,\C),H^2(\C_+),\theta_h)\) is a maximal reflection positive Hilbert space.

Finally, we apply our results about (reflection positive) regular one-parameter groups in the context of standard subspaces and use them to provide analogues of Borchers' Theorem and the Longo–Witten Theorem to pairs \((\sV,U)\) for which the triple \((\cH,\sV,U)\) is a real regular one-parameter group or for which the quadruple \((\cH,\sV,U,J_\sV)\) is a real reflection positive regular one-parameter group respectively.

§ ACKNOWLEDGMENT

First of all, I want to thank my supervisor Prof. Dr. Karl-Hermann Neeb for the great support he gave me throughout the years. He was always there to answer my questions, point out references or to discuss new ideas. I want to thank him for accompanying me from the beginning of my studies of mathematics at the university until the end of this PhD project.

I also want to thank Tobias Simon for always being there to discuss and explain mathematical concepts and ideas that we encountered in courses or seminaries we visited together. Also a big thanks to him for proofreading this thesis.

Further, I want to thank my mother Dagmar Barbara Schober for always being there for me in the last 25 years and supporting me in everything I did, both in my personal and professional life. I am also deeply grateful to her for always spell checking my mathematical texts – including this one – even though she understands very little about mathematics.

Another big thank you goes to my father Alexander Schober who always encouraged me to proceed a scientific path. Without him I would probably not have studied math and without him I would not be at this point in life.

Finally I want to thank my wife Amy Gómez Mederos for being an immense support to me. I know that especially this last period of time of me writing this thesis demanded many sacrifices from her and I am very grateful that anyhow she supported me in every way possible.

§ NOTATION

§.§ Sets

* We set \(\gls*{R+} \coloneqq \left(0,\infty\right)\) and \(\R_{\geq 0} \coloneqq \left[0,\infty\right)\) and define \(\R_- \coloneqq -\R_+\) and \(\R_{\leq 0} \coloneqq -\R_{\geq 0}\).

* We set \(\R^\times \coloneqq \R \setminus \{0\}\) and \(\C^\times \coloneqq \C \setminus \{0\}\).

* We define the complex half planes

\begin{equation*}
\gls*{C+} \coloneqq \left\lbrace z \in \C : \mathrm{Im}\left(z\right) > 0\right\rbrace, \quad \C_r \coloneqq \left\lbrace z \in \C : \mathrm{Re}\left(z\right) > 0\right\rbrace
\end{equation*}

\begin{equation*}
\C_- \coloneqq -\C_+, \quad \C_l \coloneqq -\C_r.
\end{equation*}

* We set

\[\gls*{Torus} \coloneqq \{z \in \C: |z| = 1\} \quad \text{and} \quad \gls*{D} \coloneqq \{z \in \C: |z| < 1\}.\]

* By \(\gls*{C8}\) we denote the Riemann sphere.

* For any open subset \(U \subseteq \C_\infty\) we denote by \(\gls*{O}\left(U\right)\) the set of holomorphic functions on \(U\).

* For a set \(X\) and an involution \(\theta: X \to X\), we define

\[X^\theta \coloneqq \{x \in X: \theta x = x\}.\]

* For an algebra \(\cA\) and a subset \(E \subeq \cA\) we define the commutant \(\gls*{E'}\) by

\[E' \coloneqq \{a \in \cA: (\forall b \in E) \,ab=ba\}.\]

§.§ Hilbert spaces

* We denote scalar products in a Hilbert space by \(\gls*{ZZZbraket}\), being linear in the second argument.

* If not mentioned otherwise, we assume that all occuring Hilbert spaces are separable.

* For a Hilbert space \(\cH\) we denote by \(\gls*{BH}\) the set of bounded linear operators on \(\cH\), by \(\gls*{SH}\) the set of self-adjoint linear operators on \(\cH\) and by \(S(\cH)_+\) the set of positive linear operators on \(\cH\).

* For a Hilbert space \(\cH\) and a closed subspace \(\cK \subeq \cH\) we denote by \(\gls*{PK} \in B(\cH)\) the orthogonal projection onto \(\cK\) and by \(\gls*{pK}: \cH \to \cK\) the partial isometry with

\[p_\cK\big|_\cK = \id_\cK \quad \text{and} \quad p_\cK\big|_{\cK^\perp} = 0.\]

* For a complex Hilbert space \(\cH\) and a one-parameter (semi)group \(U: \R_{(\geq 0)} \to B(\cH)\), we write \(\gls*{DU}\) for the unbounded operator und \(\cH\) defined on

\[\cD(\partial U) \coloneqq \left\{v \in \cH: \lim_{t \to 0} \frac{1}{it}(U_t-\textbf{1})v \text{\,\,exists}\right\}\]

\[\partial U v = \lim_{t \to 0} \frac{1}{it}(U_t-\textbf{1})v\]

(cf. <cit.>).

§.§ \(\gls*{Lp}\)-spaces

* For \(1 \leq p \leq \infty\) by \(\cL^p\left(\Omega,\mathbb{K},\mu\right)\) we denote the vector space of  functions with values in the field \(\mathbb{K}=\R,\C\) on the measurable space \(\Omega\) with respect to the measure \(\mu\). Further, we write \(L^p(\Omega,\K,\mu)\) for the corresponding Banach space of equivalence classes of such functions with respect to the equivalence relation given by

\[f \sim g \qquad :\Leftrightarrow \qquad \left(\exists E \subeq \Omega\right) : \left(\mu(E) = 0 \quad \text{and} \quad \forall x \in \Omega \setminus E : f(x) = g(x)\right).\]

* For \(n \in \N\), we denote the Lebesgue-measure on \(\R^n\) by \(\gls*{ln}\).

* For \(\Omega \subseteq \R^n\) and \(1 \leq p \leq \infty\) we set

\[L^p(\Omega,\K) \coloneqq L^p\left(\Omega,\K,\lambda_n\big|_\Omega\right).\]

* For a Hilbert space \(\cH\) over a field \(\mathbb{K}=\R,\C\), we define the Hilbert space

\[L^2(\Omega,\cH,\mu) \coloneqq L^2(\Omega,\mathbb{K},\mu) \,\hat{\otimes}\, \cH.\]

Interpreting \(L^2(\Omega,\cH,\mu)\) as \(\cH\)-valued, almost everywhere defined functions on \(\R\) with

\[(f \otimes v)(x) = f(x) \cdot v, \quad f \in L^2(\Omega,\mathbb{K},\mu), v \in \cH, x \in \R,\]

the scalar product on \(L^2(\Omega,\cH,\mu)\) is given by

\[\braket*{f}{g} \coloneqq \int_\Omega \braket*{f(x)}{g(x)}_\cH\,dx.\]

* For a Hilbert space \(\cH\) over a field \(\mathbb{K} = \R,\C\) and \(g \in L^\infty(\Omega,\mathbb{K},\mu)\), we define the multiplication operator \(\gls*{Mg} \in B(L^2(\Omega,\cH,\mu))\) by

\[(M_g f)(x) = g(x) \cdot f(x), \quad x \in \R.\]

* Let \(\cH\) be a Hilbert space over a field \(\mathbb{K}=\R,\C\) and let \((\Omega,\mu)\) be a \(\sigma\)-finite measure space. We call a function \(f: \Omega \to B(\cH)\) weak-\(*\)-measurable if, for every trace-class operator \(T \in B_1(\cH)\), the function

\[f_T: \Omega \to \K, \quad x \mapsto \mathrm{Tr}(Tf(x))\]

is measurable. We write \(\cL^\infty(\Omega,B(\cH),\mu)\) for the set of bounded weak-\(*\)-measurable functions \(f: \Omega \to B(\cH)\). Further, we write \(L^\infty(\Omega,B(\cH),\mu)\) for the corresponding Banach space of equivalence classes of such functions with respect to the equivalence relation given by

\[f \sim g \qquad :\Leftrightarrow \qquad \left(\exists E \subeq \Omega\right) : \left(\mu(E) = 0 \quad \text{and} \quad \forall x \in \Omega \setminus E : f(x) = g(x)\right).\]

For \(g \in L^\infty(\Omega,B(\cH),\mu)\) we define the multiplication operator \(M_g \in B(L^2(\Omega,\cH,\mu))\) by

\[(M_g f)(x) = g(x) \cdot f(x), \quad x \in \R.\]

§.§ Complexification

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

* We denote the complexification of \(\cH\) by

\[\cH_\C \coloneqq \C \otimes_\R \cH.\]

* By \(\cC_\cH\) we denote the complex conjugation

\[\gls*{CH}: \cH_\C \to \cH_\C, \quad z \otimes v \mapsto \overline{z} \otimes v.\]

* We will identify \(\cH\) with the real subspace \(\cH_\C^{\cC_\cH} = \R \otimes_\R \cH\). Under this identification, we have \(\cH_\C = \cH + i\cH\).

* For an operator \(A \in B(\cH)\), by \(\gls*{AC} \in B(\cH_\C)\), we denote the operator with

\[A_\C(x+iy) = Ax + iAy \quad \forall x,y \in \cH.\]

§.§ Tensor products

* For two Hilbert spaces \(\cH,\cK\) we define the tensor product Hilbert space

\[\cH \,\hat{\gls*{ZZZtp}}\, \cK\]

as the completion of the vector space \(\cH \otimes \cK\) with respect to the sesquilinear map on \(\cH \otimes \cK\) defined by

\[\braket*{a \otimes b}{v \otimes w} \coloneqq \braket*{a}{v}_\cH \cdot \braket*{b}{w}_\cK,\]

where \(\braket*{\cdot}{\cdot}_\cH\) denotes the scalar product on \(\cH\) and \(\braket*{\cdot}{\cdot}_\cK\) the scalar product on \(\cK\) respectively (cf. <cit.>, <cit.>).

* For two Hilbert spaces \(\cH,\cK\) we consider the inclusion

\[\iota: B(\cH) \otimes B(\cK) \to B(\cH \,\hat{\otimes}\, \cK)\]

\[\iota(A \otimes B)(v \otimes w) = (Av) \otimes (Bw)\]

(cf. <cit.>, <cit.>). Now, given two von Neumann algebras \(\cM \subeq B(\cH)\) and \(\cN \subeq B(\cK)\), we define their tensor product by

\[\cM \overline{\otimes} \cN \coloneqq \iota(\cM \otimes \cN)''.\]

We endow \(B(\cH) \otimes B(\cK)\) with the norm

\[\left\lVert \cdot \right\rVert_{B(\cH \,\hat{\otimes}\, \cK)} \circ \iota: B(\cH) \otimes B(\cK) \to \R_{\geq 0},\]

where \(\left\lVert \cdot \right\rVert_{B(\cH \,\hat{\otimes}\, \cK)}\) denotes the operator norm on \(B(\cH \,\hat{\otimes}\, \cK)\) and define

\[B(\cH) \,\hat{\otimes}\, B(\cK)\]

as the completion of the vector space \(B(\cH) \otimes B(\cK)\) with respect to this norm. Further, for two subsets \(M \subeq B(\cH)\) and \(N \subeq B(\cK)\) we define the tensor product

\[M \,\hat{\otimes}\, N \coloneqq \overline{M \otimes N} \subeq B(\cH) \,\hat{\otimes}\, B(\cK).\]

§ INTRODUCTION

In algebraic quantum field theory (AQFT) objects of current interest are standard subspaces, i.e. real subspaces \(\sV \subeq \cH\) of a complex Hilbert space \(\cH\) that satisfy

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

(cf. [21], [18], [26], [27]). Given a standard subspace \(\sV \subeq \cH\), one has the densely defined Tomita operator

\[T_\sV : \sV+i\sV \to \cH, \quad x+iy \mapsto x-iy\]

as well as the anti-unitary involution \(J_\sV\) and the positive densely defined operator \(\Delta_\sV\) that one obtains by the polar decomposition

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

of the Tomita operator. This pair \((\Delta_\sV,J_\sV)\) is referred to as the pair of modular objects of \(\sV\). In the context of standard subspaces one often studies unitary one-parameter groups on \(\cH\) whose semigroup in positive direction consists of unitary operators that map the standard subspace \(\sV\) into itself, i.e. unitary one-parameter groups \(U: \R \to \U(\cH)\) with

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

If the self-adjoint infinitesimal generator

\[\partial U \coloneqq \lim_{t \to 0} \frac{1}{it}(U_t-\textbf{1})\]

of such a unitary one-parameter group \(U\) additionally is a positive operator with trivial kernel, i.e.

\[\partial U \geq 0 \qquad \text{and} \qquad \ker (\partial U) = \{0\},\]

one calls \((\sV,U)\) a non-degenerate standard pair on \(\cH\). Two classical results about these non-degenerate standard pairs are given by Borchers' Theorem (cf. thm:Borchers, <cit.>, <cit.>) and the Longo–Witten Theorem (cf. <cit.>). The theorem by Borchers states that, given a standard pair \((\sV,U)\) on a Hilbert space \(\cH\), one can, using \(U\) and the modular data \((\Delta_\sV,J_\sV)\), construct a so called strictly positive energy representation (cf. def:PosEnergyRep) of the affine group \(\mathrm{Aff}(\R)\) on the Hilbert space \(\cH\). By the representation theory of \(\mathrm{Aff}(\R)\) one then knows that there exists a real Hilbert space \(\cK\) and a unitary operator

\[\psi: \cH \to L^2(\R,\cK_\C)^\sharp\]

\[\psi(\sV) = H^2(\C_+,\cK_\C)^\sharp \qquad \text{and} \qquad \psi \circ U_t \circ \psi^{-1} = S_t \quad \forall t \in \R.\]

Here \(L^2(\R,\cK_\C)^\sharp\) denotes the Hilbert space of square integrable \(\cK_\C\)-valued functions one \(\R\) that are invariant under the involution \(\sharp\) defined by

\[f^\sharp(x) = \overline{f(-x)}, \quad x \in \R\]

and by \(H^2(\C_+,\cK_\C)^\sharp\) we denote the corresponding Hardy space, i.e. the subspace of \(L^2(\R,\cK_\C)^\sharp\) whose Fourier transform vanishes on the negative half-line. Further \(S_t\) denotes the unitary operator on \(L^2(\R,\cK_\C)^\sharp\) defined by

\[(S_t f)(x) \coloneqq e^{itx} f(x), \quad x \in \R.\]

Under the identification

\[\cH \cong L^2(\R,\cK_\C)^\sharp \qquad \text{and} \qquad \sV \cong H^2(\C_+,\cK_\C)^\sharp\]

the theorem by Longo and Witten then states that the semigroup

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

is precisely given by the semigroup \(\Inn(\C_+,B(\cK_\C))^\sharp\) of symmetric inner functions on the upper half-plane (cf. def:Inner).

Since also unitary representations with non-positive infinitesimal generator are a subject of investigation in AQFT (cf. [22]), in this thesis we want to provide theorems similiar to the ones of Borchers and of Longo–Witten for a more general framework of unitary one-parameter groups \(U: \R \to \U(\cH)\) with

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

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 – there exists an isomorhpism between \((\cH,\sV,U)\) and \((L^2(\R,\cK_\C)^\sharp,H^2(\C_+,\cK_\C)^\sharp,S)\) for some real Hilbert space \(\cK\). By the Lax–Phillips Theorem (thm:LaxPhillipsReal,<cit.>) this condition is equivalent to \((\cH,\sV,U)\) being a so called real regular one-parameter group.

In Chapter 2 we will introduce regular one-parameter groups, i.e. triples \((\cE,\cE_+,U)\) of a complex Hilbert space \(\cE\), a closed subspace \(\cE_+ \subeq \cE\) and a unitary one-parameter group \(U: \R \to \U(\cE)\) that satisfy

\[U_t \cE_+ \subeq \cE_+ \quad \forall t \geq 0, \qquad \bigcap_{t \in \R} U_t \cE_+ = \{0\} \qquad \text{and} \qquad \overline{\bigcup_{t \in \R} U_t \cE_+} = \cE.\]

By the Lax–Phillips Theorem (thm:LaxPhillipsReal, thm:LaxPhillipsComplex) these regular one-parameter groups are always unitarily equivalent to \((L^2(\R,\cK_\C)^\sharp,H^2(\C_+,\cK_\C)^\sharp,S)\) for some real Hilbert space \(\cK\) if \(\cE\) is a real Hilbert space or to \((L^2(\R,\cK),H^2(\C_+,\cK),S)\) for some complex Hilbert space \(\cK\) if \(\cE\) is a complex Hilbert space. In this chapter we show that the so obtained multiplicity space \(\cK\) is unique up to isomorphism (cf. thm:kernelAdjointGeneral, thm:kernelAdjointGeneralReal) and that – up to unitary equivalence – the space \(\cE_+\) is unique, i.e. if \((\cE,\cE_+,U)\) and \((\cE,\cE_+',U)\) are both regular one-parameter groups, then

\[(\cE,\cE_+,U) \cong (\cE,\cE_+',U)\]

via some isomorphism of regular one-parameter groups (cf. thm:IndependentFromSubspace).

In Chapter 3 we study the semigroup \(\Inn(\C_+,B(\cH))\) of inner functions appearing in the context of the Longo–Witten Theorem. One-parameter semigroups in \(\Inn(\C_+,B(\cH))\) are infinitesimally generated by operator-valued Pick functions (cf. def:Pick), i.e. they are of the form

\[U^F_t \coloneqq e^{itF}, \quad t \in \R,\]

for some Pick function \(F\) with values in \(B(\cH)\). For \(\dim \cH < \infty\) we give criteria on the function \(F\) that are equivalent to \((L^2(\R,\cH),H^2(\C_+,\cH),U^F)\) being a regular one-parameter group (cf. thm:SpecNonDeg, thm:SpecNonDegZeroSet). In these cases we are also interested in calculating the multiplicity space \(\cK_F\) of the regular one-parameter group \((L^2(\R,\cH),H^2(\C_+,\cH),U^F)\). A formula for this multiplicity space \(\cK_F\) is provided in thm:BigLWTheorem. An interesting open problem in this context is whether similar results also hold in the case of a infinite dimensional Hilbert space \(\cH\).

Finally, we will prove that the dimension of the multiplicity space is compatible with composition of Pick functions: Given Pick functions \(f\) and \(g\) with values in \(\C\) and a Pick function \(F\) with values in \(B(\cH)\) such that \((L^2(\R,\C),H^2(\C_+),U^f)\), \((L^2(\R,\C),H^2(\C_+),U^g)\) and \((L^2(\R,\cH),H^2(\C_+,\cH),U^F)\) are regular one-parameter groups, the function \(f \circ F \circ g\) is also a Pick function such that the triple \((L^2(\R,\cH),H^2(\C_+,\cH),U^{f \circ F \circ g})\) is a regular one-parameter group and for its multiplicity space \(\cK_{f \circ F \circ g}\) one has

\[\dim \cK_{f \circ F \circ g} = \dim \cK_f \cdot \dim \cK_F \cdot \dim \cK_g\]

(cf. cor:CompNonDeg, thm:InvariantHomo).

In Chapter 4 we add another structure to regular one-parameter groups that corresponds to the involution \(J_\sV\) in the context of standard subspaces. Concretely we consider so called reflection positive regular one-parameter groups. These are quadruples \((\cE,\cE_+,U,\theta)\), where \((\cE,\cE_+,U)\) is a regular one-parameter group and \(\theta\) is an involution on \(\cE\) with

\[\theta U_t = U_{-t}\theta \quad \forall t \in \R\]

such that \((\cE,\cE_+,\theta)\) is a reflection positive Hilbert space, i.e.

\begin{equation*}
\braket*{\xi}{\theta\xi} \geq 0 \qquad \forall \xi \in \mathcal{E}_+
\end{equation*}

(cf. def:RPHS). This will lead us to the question for which functions \(h \in L^\infty(\R,\U(\cK))^\flat\) the triple \((L^2(\R,\cK),H^2(\C_+,\cK),\theta_h)\) is a reflection positive Hilbert space, where \(L^\infty(\R,\U(\cK))^\flat\) denotes the set of essentially bounded measurable functions on \(\R\) with values in \(\U(\cK)\) that are invariant under the involution \(\flat\) defined by

\[f^\flat(x) \coloneqq f(-x)^*, \quad x \in \R\]

and by \(\theta_h\) we denote the involution on \(L^2(\R,\cK)\) defined by

\[(\theta_h f)(x) = h(x) \cdot f(-x), \quad x \in \R.\]

This problem has already been described in <cit.> and is in full generality still open and should be subject of further investigation. However we were able to classify all such functions \(h\) under the two additional assumptions that \(\cK = \R\) and that \((L^2(\R,\R),H^2(\C_+),\theta_h)\) is a maximal reflection positive Hilbert space. The maximality here is to be understood in the way that, for every subspace \(\cE_+ \supeq H^2(\C_+)\) for which \((L^2(\R,\R),\cE_+,\theta_h)\) is a reflection positive Hilbert space, one has \(\cE_+ = H^2(\C_+)\) (cf. def:maxRPHS). A full characterization of the functions \(h \in L^\infty(\R,\T)^\flat\) for which the triple \((L^2(\R,\R),H^2(\C_+),\theta_h)\) is a maximal reflection positive Hilbert space is given in thm:GeneralizedMaxPosForm.

Finally, given a function \(h \in L^\infty(\R,\T)^\flat\) for which the triple \((L^2(\R,\R),H^2(\C_+),\theta_h)\) is a maximal reflection positive Hilbert space, in cor:OSTrafoFromMeasures we give an explicit description of the Osterwalder–Schrader transform (cf. <cit.>, def:OSTrafo) of this reflection positive Hilbert space. Here, given a reflection positive Hilbert space \((\cE,\cE_+,\theta)\) and defining

\[\cN_\theta \coloneqq \{\eta \in \cE_+: \braket*{\eta}{\theta \eta} = 0\},\]

the Osterwalder–Schrader transform \(\hat \cE\) of \(\cE_+\) is defined as the Hilbert space completion of \(\cE_+ \slash \cN_\theta\) with respect to the scalar product

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

In Chapter 5 we apply our results from Chapter 4 to Hankel operators, i.e. bounded linear operators \({H \in B(H^2(\C_+))}\) that satisfy

\[S_t^* H = H S_t \quad \forall t \in \R_+\]

(cf. def:Hankel), where

\[(S_t f)(x) = e^{itx} f(x), \quad f \in H^2(\C_+), x \in \R, t \in \R_+.\]

We will see that, given a 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 reflection positive Hilbert space, denoting by \(p_+\) the partial isometry from \(L^2(\R,\C)\) onto its subspace \(H^2(\C_+)\), the operator

\[H_h \coloneqq p_+ \theta_h p_+^*\]

is a positive Hankel operator with \({\left\lVert H_h\right\rVert \leq 1}\). The question we investigate in Chapter 5, which has already been under investigation in [2], is if every positive Hankel operator \(H\) with \(\left\lVert H\right\rVert \leq 1\) is of this form. In <cit.> it has already been shown that this is true if one allows to change the scalar product on \(H^2(\C_+)\). We prove that in fact the statement is true without changing the scalar product (cf. thm:HankelHasUnimodMeasure).

In Chapter 6 we apply the results of the previous chapters in the context of standard subspaces. Given a complex Hilbert space \(\cH\), a standard subspace \(\sV \subeq \cH\) and a strongly continuous one-parameter group \(U: \R \to \U(\cH)\), we consider regular pairs, i.e. pairs \((\sV,U)\), for which \((\cH,\sV,U)\) is a real regular one-parameter group and reflection positive regular pairs, i.e. pairs \((\sV,U)\), for which \((\cH,\sV,U,J_\sV)\) is a real reflection positive regular one-parameter group (cf. def:RegPair). 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 (cf. cor:standardPairIsRPRegPair). We will also show that in both cases the triple \((\cH,\sV,J_\sV)\) is a maximal reflection positive Hilbert space (RPHS) (cf. prop:StandardSubspaceMaxPos).

By the Lax–Phillips Theorem we know that, for every regular pair \((\sV,U)\) on a Hilbert space \(\cH\), the regular one-parameter group \((\cH,\sV,U)\) is equivalent to \((L^2(\R,\cK_\C)^\sharp,H^2(\C_+,\cK_\C)^\sharp,S)\) for some real Hilbert space \(\cK\). Here the multiplicity free case \(\cK = \R\) corresponds to the representation \(U\) being \(\R\)-cyclic, i.e.

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

(cf. prop:MultiplicityRCyclic).

In thm:LWExtended we show that, for all regular pairs \((\sV,U)\), identifying

\[\cH \cong \cH_\cK \coloneqq L^2(\R,\cK_\C)^\sharp \qquad \text{and} \quad \sV \cong \sV_\cK \coloneqq H^2(\C_+,\cK_\C)^\sharp,\]

the semigroup

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

is given by the semigroup \(\Inn(\C_+,B(\cK_\C))^\sharp\) of symmetric inner functions and thereby provide a generalization of the Longo–Witten Theorem (cf. <cit.>). In this theorem we also provide some conditions under which semigroups in \(\cE(\sV,U)\) extend to regular regular one-parameter groups by using the results from Chapter 3.
Many of our main results are summarized in the following table:

24ptM\((\sV,U)\)-24ptM     regular pair     \(\substack{\text{\normalsize \bf reflection positive} \\ \text{\normalsize \bf regular pair}}\)     \(\substack{\text{\normalsize \bf non-degenerate} \\ \text{\normalsize \bf standard pair}}\)

24ptM Group \(G\)-24ptM     \(\R\)     \(\R \rtimes \{-1,1\}\)     \(\mathrm{Aff}(\R) = \R \rtimes \R^\times\)

24ptM\(\substack{\text{\normalsize \bf Representation} \\ \text{\normalsize \boldmath\(\rho: G \to \mathrm{AU}(\cH)\)}}\)-24ptM      \(\rho(t) = U_t\)     \(\substack{\text{\normalsize \(\rho(t,1) = U_t\)} \\ \text{\normalsize \(\rho(0,-1) = J_\sV\)}}\)     \(\substack{\text{\normalsize \(\rho(t,1) = U_t\)} \\ \text{\normalsize \(\rho(0,e^s) = \Delta^{-\frac{is}{2\pi}}\)} \\ \text{\normalsize \(\rho(0,-1) = J_\sV\)}}\)

24ptM\(\substack{\text{\normalsize \bf Normal form} \\ \text{\normalsize \bf for \boldmath\((\cH,\sV,U,J_\sV)\)}}\) -24ptM     \(\substack{\text{\normalsize \((\cH_\cK,\sV_\cK,S,\theta)\) with} \\ \text{\normalsize \((\cH_\cK,\sV_\cK,\theta)\) being a} \\ \text{\normalsize maximal RPHS}}\)     \(\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}}\)     \((\cH_\cK,\sV_\cK,S,\theta_{i \cdot \sgn})\)

24ptM\(\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}}\) -24ptM     \(\substack{\text{\normalsize \((\cH_\R,\sV_\R,S,\theta)\) with} \\ \text{\normalsize \((\cH_\R,\sV_\R,\theta)\) being a} \\ \text{\normalsize maximal RPHS}}\)     \(\substack{\text{\normalsize \((\cH_\R,\sV_\R,S,\theta_{h_\nu})\) with} \\ \text{\normalsize \(\nu \in \mathcal{BM}^{\mathrm{fin}}([0,\infty]) \setminus \{0\}\)}}\)     \(\substack{\text{\normalsize \((\cH_\R,\sV_\R,S,\theta_{i \cdot \sgn}) =\)} \\ \text{\normalsize \((\cH_\R,\sV_\R,S,\theta_\nu)\) with} \\ \text{\normalsize \(d\nu(\lambda) = \frac 2{1+\lambda^2}\)}}\)

24ptM\(\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}}\)-24ptM     \(\overline{\sV_\R \slash \cN_\theta}\)     \(L^2(\R_+,\C,\cT \nu)\)     \(L^2(\R_+,\C,2\lambda_1)\)

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})}}\)

§ REGULAR ONE-PARAMETER GROUPS

In this chapter we will introduce the concept of a regular one-parameter group (ROPG). Using the Lax–Phillips Theorem, we will introduce a standard form for these ROPGs and later we will provide some basic theory about ROPGs.

§.§ ROPGs and the Lax–Phillips Theorem

Consider a triple \((\cE,\cE_+,U)\) of a complex Hilbert space \(\cE\), a closed subspace \(\cE_+ \subeq \cE\) and a unitary one-parameter group \(U: \R \to \U(\cE)\).

* We call \((\cE,\cE_+,U)\) a semi-regular one-parameter group (*SROPG), if

\[U_t \cE_+ \subeq \cE_+ \quad \forall t \geq 0 \qquad \text{and} \qquad \bigcap_{t \in \R} U_t \cE_+ = \{0\}.\]

* We define the unitary one-parameter group \(\check U: \R \to \U(\cE)\) by \(\check U_t = U_{-t}\), \(t \in \R\).

* We call \((\cE,\cE_+,U)\) a regular one-parameter group (*ROPG), if \((\cE,\cE_+,U)\) and \((\cE,\cE_+^\perp,\check U)\) are both a SROPG.

* We call a (S)ROPG \((\cE,\cE_+,U)\) real/complex, if \(\cE\) is real/complex.

* We say that two ROPGs \((\cE,\cE_+,U)\) and \((\cE',\cE_+',U')\) are equivalent, if there exists a unitary operator \(\psi: \cE \to \cE'\) such that

\[\psi(\cE_+) = \cE_+' \quad \text{and} \quad \psi \circ U_t = U_t' \circ \psi \quad \forall t \in \R.\]

We call such an operator \(\psi\) an isomorphism between \((\cE,\cE_+,U)\) and \((\cE',\cE_+',U')\).

For \(t \geq 0\) one has \(U_t \cE_+ \subeq \cE_+\), if and only if

\[\cE_+^\perp \subeq (U_t \cE_+)^\perp = U_t \cE_+^\perp = \check U_{-t} \cE_+^\perp,\]

which is equivalent to \(\check U_t \cE_+^\perp \subeq \cE_+^\perp\). Further

\[\left(\bigcap_{t \in \R} \check U_t \cE_+^\perp\right)^\perp = \overline{\bigcup_{t \in \R} \left(\check U_t \cE_+^\perp\right)^\perp} = \overline{\bigcup_{t \in \R} \check U_t \cE_+} = \overline{\bigcup_{t \in \R} U_t \cE_+}.\]

Therefore \((\cE,\cE_+,U)\) is a ROPG, if and only if

\[U_t \cE_+ \subeq \cE_+ \quad \forall t \geq 0\]

\[\bigcap_{t \in \R} U_t \cE_+ = \{0\} \qquad \text{and} \qquad \overline{\bigcup_{t \in \R} U_t \cE_+} = \cE.\]

The following theorem gives us a normal form for ROPGs:
(Lax–Phillips)(cf. <cit.><cit.>, <cit.>)
For every real/complex Hilbert space \(\cK\) the triple \((L^2(\R,\cK),L^2(\R_+,\cK),\tau)\) is a real/complex ROPG, where the one-parameter group \(\gls*{tau}: \R \to U(L^2(\R,\cK))\) is the translation action defined by

\[(\tau_tf)(p) = f(p-t), \qquad t,p \in \R.\]

Further, every real/complex ROPG \((\cE,\cE_+,U)\) is equivalent to \((L^2(\R,\cK),L^2(\R_+,\cK),\tau)\) for some real/complex Hilbert space \(\cK\).

In the notation used in the theorem we identify \(L^2(\R_+,\cK)\) with the subspace

\[\left\{f \in L^2(\R,\cK): f\big|_{(-\infty,0]}\equiv 0\right\} \subeq L^2(\R,\cK).\]

The version of the Lax–Phillips Theorem, that we will use in the following, is slightly different, since we will change to the Fourier transformed picture. For this we need some definitions:

Let \(\cK\) be a real Hilbert space and \(\cC_\cK\) be the complex conjugation on \(\cK_\C\).

* In the following we identify \(L^2(\R,\cK)\) with the subspace

\[\left\{f \in L^2(\R,\cK_\C): \left(\forall x \in \R\right) \, f(x) \in \cK_\C^{\cC_\cK} \right\} \subeq L^2(\R,\cK_\C).\]

* By \(\sharp: L^2(\R,\cK_\C) \to L^2(\R,\cK_\C)\), we denote the involution defined by

\[\gls*{fsharp}(x) = \cC_\cK f(-x).\]

* For a complex Hilbert space \(\cH\) we consider the Fourier transform \(\gls*{fourier}: L^2(\R,\cH) \to L^2(\R,\cH)\) given by

\[(\cF f)(p) \coloneqq \frac 1{\sqrt{2\pi}} \int_\R e^{-ipx}f(x) \,dx\]

for every \(f \in L^1(\R,\C) \cap L^2(\R,\C)\).

To change to the Fourier transformed picture, we first notice that

\[(\cF^{-1} \tau_t f)(x) = \frac 1{\sqrt{2\pi}} \int_\R e^{ipx}f(p-t) \,dp = \frac 1{\sqrt{2\pi}} \int_\R e^{i(p+t)x}f(x) \,dp = e^{itx} (\cF^{-1} f)(x).\]

Further, for a real Hilbert space \(\cK\), we have

\[(\cF^{-1} \cC_\cK f)(x) = \frac 1{\sqrt{2\pi}} \int_\R e^{ipx}\cC_\cK f(p) \,dp = \cC_\cK \frac 1{\sqrt{2\pi}} \int_\R e^{-ipx}f(p) \,dp = (\cF^{-1} f)^\sharp(x),\]

\[\cF^{-1}L^2(\R,\cK) = L^2(\R,\cK_\C)^\sharp.\]

Further, for a complex Hilbert space \(\cH\), by <cit.>, we have

\[\cF^{-1}L^2(\R_+,\cH) = H^2(\C_+,\cH),\]

where \(H^2(\C_+,\cH) \subeq L^2(\R,\cH)\) denotes the \(\cH\)-valued Hardy space (cf. sec:HardySpaces).

With this information one immediately gets a version of the Lax–Phillips Theorem in the Fourier transformed picture:
(Lax–Phillips, complex Fourier version)
For every complex Hilbert space \(\cK\) the triple \((L^2(\R,\cK),H^2(\C_+,\cK),S)\) with the one-parameter group

\[\gls*{S}: \R \to U(L^2(\R,\cK)), \quad (S_tf)(x) = e^{itx}f(x)\]

is a complex ROPG and every complex ROPG \((\cE,\cE_+,U)\) is equivalent to \((L^2(\R,\cK),H^2(\C_+,\cK),S)\) for some complex Hilbert space \(\cK\).

(Lax–Phillips, real Fourier version)
For every real Hilbert space \(\cK\) the triple \((L^2(\R,\cK_\C)^\sharp,H^2(\C_+,\cK_\C)^\sharp,S)\) with the one-parameter group

\[S: \R \to U(L^2(\R,\cK_\C)^\sharp), \quad (S_tf)(x) = e^{itx}f(x)\]

is a real ROPG and every real ROPG \((\cE,\cE_+,U)\) is equivalent to \((L^2(\R,\cK_\C)^\sharp,H^2(\C_+,\cK_\C)^\sharp,S)\) for some real Hilbert space \(\cK\).

The one-parameter group \(S\) appearing in the last two theorems, will appear often in the following sections. Many times an object of interest will be its commutant \(S(\R)'\). To compute this commutant, we need the following theorem:
(cf. <cit.>)
Let \(\cH,\cK\) be Hilbert spaces and let \(M \subeq B(\cH)\) and \(N \subeq B(\cK)\) be von Neumann algebras. Then

\[(M \,\overline{\otimes}\, N)' = M' \,\overline{\otimes}\, N'.\]

Let \(\cK\) be a complex Hilbert space and \(S: \R \to U(L^2(\R,\cK))\) be the one-parameter group defined by

\[(S_tf)(x) = e^{itx}f(x).\]

\[S(\R)' = M\left(L^\infty(\R,B(\cK))\right).\]

Considering the functions \(e_t \in L^\infty(\R,\C)\) with \(e_t(x) = e^{itx}\) and identifying

\[L^2(\R,\cK) \cong L^2(\R,\C) \,\hat{\otimes}\, \cK,\]

we have that

\[S_t = M_{e_t} \otimes \textbf{1}_\cK, \quad t \in \R.\]

Then, by prop:etDense, the set \(S(\R)\) spans a weakly dense subspace in \(M(L^\infty(\R,\C)) \,\overline{\otimes}\, \C\textbf{1}_\cK\). This, by thm:CommutantTensor, yields

\begin{align*}
S(\R)' &= \left(M(L^\infty(\R,\C))  \,\overline{\otimes}\, \C\textbf{1}_\cK\right)' = M(L^\infty(\R,\C))'  \,\overline{\otimes}\, (\C\textbf{1}_\cK)' = M(L^\infty(\R,\C))  \,\overline{\otimes}\, B(\cK),
\end{align*}

using <cit.> in the last step. The statement now follows since, by <cit.>, we have

\[M(L^\infty(\R,\C)) \,\overline{\otimes}\, B(\cK) = M\left(L^\infty(\R,B(\cK))\right). \qedhere\]

§.§ Formulas for the multiplicity space

In this section, given a ROPG \((\cE,\cE_+,U)\), we will provide some ways to calculate the space \(\cK\) appearing in the Lax–Phillips Theorem, thereby showing that this space \(\cK\) is unique up to isomorphism.

Let \((\cE,\cE_+,U)\) be a complex ROPG and \(\cK\) a complex Hilbert space such that \((\cE,\cE_+,U)\) is equivalent to \((L^2(\R,\cK),H^2(\C_+,\cK),S)\). Then

\[B(\cK) \cong (U(\R) \cup \{P_{\cE_+}\})'.\]

Without loss of generality we can assume that \((\cE,\cE_+,U) = (L^2(\R,\cK),H^2(\C_+,\cK),S)\). Let \(A \in (S(\R) \cup \{P_{H^2(\C_+,\cK)}\})'\). Then \(A \in S(\R)'\), by prop:SCommutant, implies that \(A=M_f\) with \(f \in L^\infty(\R,B(\cK))\). Further \({A \in \{P_{H^2(\C_+,\cK)}\}'}\) implies

\[M_f H^2(\C_+,\cK) \subeq H^2(\C_+,\cK) \quad \text{and} \quad M_f H^2(\C_-,\cK) \subeq H^2(\C_-,\cK),\]

which, by prop:H2InclusionFunctions, yields

\[f \in H^\infty(\C_+,B(\cK)) \quad \text{and} \quad f \in H^\infty(\C_-,B(\cK)),\]

so \(f = C \cdot \textbf{1}\) for some \(C \in B(\cK)\) by cor:HInftyIntersection.

Conversely, for every \(C \in B(\cK)\) one has \(M_{C \cdot \textbf{1}} \in (S(\R) \cup \{P_{H^2(\C_+,\cK)}\})'\), so

\[(S(\R) \cup \{P_{H^2(\C_+,\cK)}\})' = B(\cK) \cdot M_\textbf{1}. \qedhere\]

Let \((\cE,\cE_+,U)\) be a real ROPG and \(\cK\) a real Hilbert space such that \((\cE,\cE_+,U)\) is equivalent to \((L^2(\R,\cK_\C)^\sharp,H^2(\C_+,\cK_\C)^\sharp,S)\). Then

\[B(\cK)^2 \cong (U(\R) \cup \{P_{\cE_+}\})'.\]

Without loss of generality we can assume that \((\cE,\cE_+,U) = (L^2(\R,\cK_\C)^\sharp,H^2(\C_+,\cK_\C)^\sharp,S)\). Let \(A \in (S(\R) \cup \{P_{H^2(\C_+,\cK_\C)^\sharp}\})'\). Then \(A \in S(\R)'\), by prop:SCommutant, implies that \(A=M_f\) with \({f \in L^\infty(\R,B(\cK_\C))}\). Further \({A \in \{P_{H^2(\C_+,\cK_\C)^\sharp}\}'}\) implies

\[M_f H^2(\C_+,\cK_\C)^\sharp \subeq H^2(\C_+,\cK_\C)^\sharp \quad \text{and} \quad M_f H^2(\C_-,\cK_\C)^\sharp \subeq H^2(\C_-,\cK_\C)^\sharp.\]

Since \(H^2(\C_+,\cK_\C) = H^2(\C_+,\cK_\C)^\sharp + i H^2(\C_+,\cK_\C)^\sharp\), this implies

\[M_f H^2(\C_+,\cK_\C) \subeq H^2(\C_+,\cK_\C) \quad \text{and} \quad M_f H^2(\C_-,\cK_\C) \subeq H^2(\C_-,\cK_\C),\]

which, by prop:H2InclusionFunctions and cor:HInftyIntersection, yields \(f = C \cdot \textbf{1}\) for some \(C \in B(\cK_\C)\). Further \(AL^2(\R,\cK_\C)^\sharp \subeq L^2(\R,\cK_\C)^\sharp\) implies that \([\cC_\cK,C] = 0\), so \(C \in \{\cC_\cK\}'\). Conversely, for every \(C \in \{\cC_\cK\}'\) one has \(M_{C \cdot \textbf{1}} \in (S(\R) \cup \{P_{H^2(\C_+,\cK_\C)^\sharp}\})'\), so

\[(S(\R) \cup \{P_{H^2(\C_+,\cK_\C)^\sharp}\})' = \{\cC_\cK\}' \cdot M_\textbf{1}.\]

The statement then follows by the fact that

\[\Phi: B(\cK)^2 \to \{\cC_\cK\}', \quad \Phi(A,B)(x+iy) = Ax+iBy, \quad A,B \in B(\cK), \quad x,y \in \cK\]

defines an isomorphism.

Let \(\cK\) be a real Hilbert space and \(\lambda \in \C_+\). Then

\[\left(Q_\lambda \cdot \cK_\C\right) \cap H^2(\C_+,\cK_\C)^\sharp = \begin{cases} Q_\lambda \cdot \cK & \text{if } \lambda \in i\R_+ \\ \{0\} & \text{if } \lambda \in \C_+ \setminus i\R_+, \end{cases}\]

with \(Q_\lambda \in H^2(\C_+)\) defined by

\[Q_\lambda(z) \coloneqq \frac 1{2\pi i} \frac 1{z-\overline{\lambda}}\]

(cf. prop:H2RKHS).

For \(v \in \cK_\C\), one has

\begin{align*}
(Q_\lambda \cdot v)^\sharp(z) &= \cC_\cK(Q_\lambda(-\overline{z}) \cdot v) = \overline{Q_\lambda(-\overline{z})} \cdot \cC_\cK v
\\&= \overline{\frac 1{2\pi i} \frac 1{-\overline{z}-\overline{\lambda}}} \cdot \cC_\cK v = \frac 1{2\pi i} \frac 1{z+\lambda} \cdot \cC_\cK v = Q_{-\overline{\lambda}}(z) \cdot \cC_\cK v.
\end{align*}

Therefore \((Q_\lambda \cdot v)^\sharp = Q_\lambda \cdot v\), if and only if

\[Q_{-\overline{\lambda}}(z) \cdot \cC_\cK v = Q_{\lambda}(z) \cdot v \quad \forall z \in \C_+.\]

For \(v \neq 0\) this implies that the map

\[\C_+ \ni z \mapsto \frac{Q_{\lambda}(z)}{Q_{-\overline{\lambda}}(z)} = \frac{z+\lambda}{z-\overline{\lambda}}\]

is constant, which yields \(\lambda = -\overline{\lambda}\) and therefore \(\lambda \in i\R_+\). For \(\lambda \in i\R_+\) we then have \((Q_\lambda \cdot v)^\sharp = Q_\lambda \cdot v\), if and only if \(\cC_\cK v = v\). We conclude that \(Q_\lambda \cdot v \in H^2(\C_+,\cK_\C)^\sharp\) is equivalent to \(\lambda \in i\R_+\) and \(v \in \cK_\C^{\cC_\cK} \cong \cK\).

Let \(\cK\) be a Hilbert space and set

\[\cE_+ \coloneqq \begin{cases} H^2(\C_+,\cK) & \text{if } \cK \text{ complex} \\ H^2(\C_+,\cK_\C)^\sharp & \text{if } \cK \text{ real}. \end{cases}\]

Further, let

\[A \coloneqq p_{\cE_+} (\partial S) p_{\cE_+}^* \in B(\cE_+).\]

Then, if \(\cK\) is a complex Hilbert space, for every \(\lambda \in \C_+\), one has

\[\ker\left(\overline{\lambda}\textbf{1}-A^*\right) = Q_\lambda \cdot \cK,\]

and if \(\cK\) is a real Hilbert space, one has

\[\ker\left(\overline{\lambda}\textbf{1}-A^*\right) = \begin{cases} Q_\lambda \cdot \cK & \text{if } \lambda \in i\R_+ \\ \{0\} & \text{if } \lambda \in \C_+ \setminus i\R_+. \end{cases}\]

We first consider the case that \(\cK\) is a complex Hilbert space. Since

\[\partial S = \lim_{t \to 0} \frac{S_t - \textbf{1}}{it} = \lim_{t \to 0} \frac{M_{e^{it \Id}} - \textbf{1}}{it} = M_{\Id},\]

for \(f \in D(A)\), we have \((Af)(z) = z f(z)\). Then, for \(\lambda \in \C_+\), \(v \in \cK\) and \(f \in D(A)\), by prop:H2RKHS, we get

\[\braket*{Q_\lambda \cdot v}{Af} = \braket*{v}{\lambda f(\lambda)}_\cK = \lambda \braket*{v}{f(\lambda)}_\cK = \lambda \braket*{Q_\lambda \cdot v}{f} = \braket*{\overline{\lambda} Q_\lambda \cdot v}{f},\]

so \(Q_\lambda \cdot v \in D(A^*)\) with \(A^*Q_\lambda \cdot v = \overline{\lambda}Q_\lambda \cdot v\) and therefore \(Q_\lambda \cdot v \in \ker\left(\overline{\lambda}\textbf{1}-A^*\right)\).

Now, conversely, let \(f \in \ker\left(\overline{\lambda}\textbf{1}-A^*\right)\). For every \(v \in \cK\) and \(a,b \in \C_+\), we have

\[(Q_a \cdot v-Q_b \cdot v)(z) = \frac{1}{2 \pi i} \left(\frac{1}{z-\overline{a}}-\frac{1}{z-\overline{b}}\right) \cdot v = \frac{1}{2 \pi i} \frac{\overline{a}-\overline{b}}{(z-\overline{a})(z-\overline{b})} \cdot v,\]

so \(Q_a \cdot v-Q_b \cdot v \in D(A)\) with

\begin{align*}
(A(Q_a \cdot v-Q_b \cdot v))(z) &= z \cdot \frac{1}{2 \pi i} \frac{\overline{a}-\overline{b}}{(z-\overline{a})(z-\overline{b})} \cdot v
\\&= \frac{1}{2 \pi i} \left(\frac{\overline{a}}{z-\overline{a}}-\frac{\overline{b}}{z-\overline{b}}\right) \cdot v = (\overline{a}Q_a \cdot v-\overline{b}Q_b \cdot v)(z).
\end{align*}

This leads to

\begin{align*}
\braket*{v}{\overline{\lambda}f(a)-\overline{\lambda}f(b)}_\cK &= \braket*{Q_a \cdot v-Q_b \cdot v}{\overline{\lambda}f} = \braket*{Q_a \cdot v-Q_b \cdot v}{A^*f}
\\&= \braket*{A(Q_a \cdot v-Q_b \cdot v)}{f} = \braket*{\overline{a}Q_a \cdot v-\overline{b}Q_b \cdot v}{f}
\\&= a\braket*{v}{f(a)}_\cK-b \braket*{v}{f(b)}_\cK = \braket*{v}{a f(a)-b f(b)}_\cK.
\end{align*}

Since this is true for every \(v \in \cK\), we get

\[(\overline{\lambda}-a)f(a) = (\overline{\lambda}-b)f(b)\]

and therefore, since \(Q_\lambda(b) = \frac 1{2\pi i} \frac 1{b-\overline{\lambda}}\), we have

\[f(b) = \frac{a-\overline{\lambda}}{b-\overline{\lambda}}f(a) = Q_\lambda(b) \cdot 2\pi i(a - \overline{\lambda})f(a) = (Q_\lambda \cdot v)(b)\]

with \(v \coloneqq 2\pi i(a - \overline{\lambda})f(a) \in \cK\). This shows that

\[\ker\left(\overline{\lambda}\textbf{1}-A^*\right) = Q_\lambda \cdot \cK.\]

Now, in the case of a real Hilbert space \(\cK\), analogously to the complex case, one shows that \(Q_\lambda \cdot v \in \ker\left(\overline{\lambda}\textbf{1}-A^*\right)\) if \(Q_\lambda \cdot v \in H^2(\C_+,\cK_\C)^\sharp\), so if \(\lambda \in i\R_+\) and \(v \in \cK\) (cf. lem:QSpaceReal). Also as in the complex case, one shows that, for every \(f \in \ker\left(\overline{\lambda}\textbf{1}-A^*\right)\), one has

\[\braket*{v}{\overline{\lambda}f(a)-\overline{\lambda}f(b)}_{\cK_\C} = \braket*{v}{a f(a)-b f(b)}_{\cK_\C}\]

for every \(a,b \in i\R_+\) and \(v \in \cK\). But since \(\cK = \C \otimes_\R \cK\), this also implies

\[\braket*{v}{\overline{\lambda}f(a)-\overline{\lambda}f(b)}_{\cK_\C} = \braket*{v}{a f(a)-b f(b)}_{\cK_\C}\]

for every \(v \in \cK_\C\) and therefore, as in the complex case, one concludes that

\[f(b) = (Q_\lambda \cdot v)(b)\]

with \(v \coloneqq 2\pi i(a - \overline{\lambda})f(a)\) for every \(b \in i\R_+\). Since both functions \(f\) and \(Q_\lambda \cdot v\) are holomorphic and coincide on \(i\R_+\), they are equal, so \(f \in Q_\lambda \cdot \cK_\C\). Using lem:QSpaceReal, this shows that

\[\ker\left(\overline{\lambda}\textbf{1}-A^*\right) = (Q_\lambda \cdot \cK_\C) \cap H^2(\C_+,\cK_\C)^\sharp = \begin{cases} Q_\lambda \cdot \cK & \text{if } \lambda \in i\R_+ \\ \{0\} & \text{if } \lambda \in \C_+ \setminus i\R_+. \end{cases} \qedhere\]

Let \((\cE,\cE_+,U)\) be a complex ROPG and \(\cK\) a complex Hilbert space such that \((\cE,\cE_+,U)\) is equivalent to \((L^2(\R,\cK),H^2(\C_+,\cK),S)\). Further let

\[A \coloneqq p_{\cE_+} (\partial U) p_{\cE_+}^* \in B(\cE_+).\]

Then, for every \(\lambda \in \C_+\), one has

\[\cK \cong \ker\left(\overline{\lambda}\textbf{1}-A^*\right).\]

This follows immediately from lem:kernelAdjoint.

Let \((\cE,\cE_+,U)\) be a real ROPG and \(\cK\) a real Hilbert space such that \((\cE,\cE_+,U)\) is equivalent to \((L^2(\R,\cK_\C)^\sharp,H^2(\C_+,\cK_\C)^\sharp,S)\). Further let

\[A \coloneqq p_{\cE_+} (\partial U) p_{\cE_+}^* \in B(\cE_+).\]

Then, for every \(\lambda \in i\R_+\), one has

\[\cK \cong \ker\left(\overline{\lambda}\textbf{1}-A^*\right).\]

This follows immediately from lem:kernelAdjoint.

Since the kernel \(\ker\left(\overline{\lambda}\textbf{1}-A^*\right)\) appearing in the last two theorems just depends on \((\cE,\cE_+,U)\) and not on the choice of \(\cK\), this theorem shows that \(\cK\) is unique up to isomorphism. We therefore can formulate the following definition:

For every complex/real ROPG \((\cE,\cE_+,U)\), by the Lax–Phillips Theorem (thm:LaxPhillipsComplex/thm:LaxPhillipsReal) and thm:kernelAdjointGeneral/thm:kernelAdjointGeneralReal, there exists an — up to isomorphism — unique complex/real Hilbert space \(\cK\) such that the ROPG \((\cE,\cE_+,U)\) is equivalent to \((L^2(\R,\cK),H^2(\C_+,\cK),S)\) or \((L^2(\R,\cK_\C)^\sharp,H^2(\C_+,\cK_\C)^\sharp,S)\) respectively. We call this Hilbert space \(\cK\) the multiplicity space of \((\cE,\cE_+,U)\).

In order to not having to prove everything separately for real and complex ROPGs, we now show how they are linked to each other:

Let \(\cE\) be a real Hilbert space, \(\cE_+ \subeq \cE\) be a closed subspace and \(U: \R \to \U(\cE)\) be a unitary one-parameter group. Then \((\cE,\cE_+,U)\) is a real ROPG, if and only if \((\cE_\C,(\cE_+)_\C,U_\C)\) is a complex ROPG. In this case, if the multiplicity space of \((\cE,\cE_+,U)\) is \(\cK\), then the multiplicity space of \((\cE_\C,(\cE_+)_\C,U_\C)\) is given by \(\cK_\C\).

For \(t \in \R\), we have

\[(U_t)_\C (\cE_+)_\C = (U_t)_\C (\cE_+ + i \cE_+) = U_t \cE_+ + i U_t\cE_+,\]

which immediately implies that \((U_t)_\C (\cE_+)_\C \subeq (\cE_+)_\C\), if and only if \(U_t \cE_+ \subeq \cE_+\). Further, we get

\[\bigcap_{t \in \R} (U_t)_\C (\cE_+)_\C = \left(\bigcap_{t \in \R} U_t \cE_+\right) + i \left(\bigcap_{t \in \R} U_t \cE_+\right)\]

\[\bigcup_{t \in \R} (U_t)_\C (\cE_+)_\C = \left(\bigcup_{t \in \R} U_t \cE_+\right) + i \left(\bigcup_{t \in \R} U_t \cE_+\right).\]

This proves that \((\cE,\cE_+,U)\) is a real ROPG, if and only if \((\cE_\C,(\cE_+)_\C,U_\C)\) is a complex ROPG.

Now let \((\cE,\cE_+,U)\) be a real ROPG with multiplicity space \(\cK\). Then there exists a isomorphism \(\psi\) of the real ROPG \((\cE,\cE_+,U)\) and \((L^2(\R,\cK_\C)^\sharp,H^2(\C_+,\cK_\C)^\sharp,S)\). Then the map \(\psi_\C\) is an isomorphism of complex ROPG. This together with the fact that

\[L^2(\R,\cK_\C) = L^2(\R,\cK_\C)^\sharp + i L^2(\R,\cK_\C)^\sharp \cong L^2(\R,\cK_\C)^\sharp_\C\]

\[H^2(\C_+,\cK_\C) = H^2(\C_+,\cK_\C)^\sharp + i H^2(\C_+,\cK_\C)^\sharp \cong H^2(\C_+,\cK_\C)^\sharp_\C\]

implies that the complex ROPG \((\cE_\C,(\cE_+)_\C,U_\C)\) is equivalent to \((L^2(\R,\cK_\C),H^2(\C_+,\cK_\C),S)\) and therefore has multiplicity space \(\cK_\C\).

In the following sections we will always first prove our results just for complex ROPGs and will later translate our final theorems also to the real case using prop:ROPGRealToComplex.

§.§ Independence of the multiplicity space from \(\cE_+\)

In this section we want to answer the question whether the multiplicity space of a ROPG \((\cE,\cE_+,U)\) depends on the subspace \(\cE_+\) when one fixes the Hilbert space \(\cE\) and the unitary one-parameter group \(U\). For this we consider our standard example \((L^2(\R,\cK),H^2(\C_+,\cK),S)\) and want to vary the space \(\cE_+\), so we consider ROPGs of the form \((L^2(\R,\cK),\cE_+,S)\) for a closed subspace \(\cE_+ \subeq L^2(\R,\cK)\) and want to calculate their multiplicity space.

The theorem by Halmos (<cit.>) that we will use to solve this problem has been formulated for the Hardy space on the unit disc \(\D\) instead of the Hardy space on the upper half plane \(\C_+\), but via Cayley transform we will be able to switch between these two pictures. To formulate this theorem by Halmos, we need some definitions:

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

* We write

\[L^2(\T,\cK) \coloneqq L^2(\T,\cK,\mu) \quad \text{and} \quad L^\infty(\T,B(\cK)) \coloneqq L^\infty(\T,B(\cK),\mu),\]

where \(\mu\) is the Haar measure on the group \(\T\) normalized by \(\mu(\T) = 1\). Further, we write \(H^2(\D,\cK) \subeq L^2(\T,\cK)\) for the corresponding Hardy space on the disc (cf. def:HardySpaceDisc).

* For \(\Omega = \T,\R\), we call \(F \in L^\infty(\Omega,\mathrm{B}(\cK))\) rigid, if there exists a subspace \(\fU \subeq \cK\) such that for almost every \(z \in \Omega\) the operator \(F(z)\) is a partial isometry with initial subspace \(\fU\).

Let \(\cH\) be a Hilbert space. We then say a closed subspace \(\fM \subeq \cH\) reduces an operator \(A \in \mathrm{B}(\cH)\) if \([A,P_\fM] = 0\). We say that a closed subspace \(\fM \subeq \cH\) is irreducible for \(A\), if no non-trivial subspace of \(\fM\) reduces \(A\).

(cf. <cit.>)
Let \(\cK\) be a complex Hilbert space and let \(\fM \subeq L^2(\T,\cK)\) be a closed subspace. Then \(\fM\) is irreducible for the operator \(M_{\Id_\T} \in \mathrm{B}\left(L^2(\T,\cK)\right)\) and satisfies \(M_{\Id_\T} \fM \subeq \fM\), if and only if there exists a rigid function \(F \in L^\infty(\T,\mathrm{B}(\cK))\) such that

\[\fM = M_F H^2(\D,\cK).\]

To use the previous theorem, we have to switch from the unit disc \(\D\) to the upper half plane \(\C_+\). We will do this by using the biholomorphic Cayley transform

\[h(z) \coloneqq \frac{z-i}{z+i}\]

that has the property that

\[h(\R) = \T \setminus \{1\} \quad \text{and} \quad h(\C_+) = \D.\]

Then \(h\) induces an isometric isomorphism of the Hilbert spaces \(L^2(\T,\cK)\) and \(L^2(\R,\cK)\) given by

\[\Gamma: L^2(\T,\cK) \to L^2(\R,\cK), \quad (\Gamma f)(x) = \frac {1}{\sqrt{\pi}(x+i)}(f \circ h)\left(x\right),\]

which which preserves the corresponding Hardy spaces, i.e.

\[\Gamma(H^2(\D,\cK)) = H^2(\C_+,\cK)\]

(cf. <cit.>). Using this map \(\Gamma\), we get the following analogue to thm:Halmos:

Let \(\cK\) be a complex Hilbert space and \(\fN \subeq L^2(\R,\cK)\) be a closed subspace such that \((L^2(\R,\cK),\fN,S)\) is a SROPG. Then there exists a rigid function \(F \in L^\infty(\R,\mathrm{B}(\cK))\) such that

\[\fN = M_F H^2(\C_+,\cK).\]

We first show that the subspace \(\fM \coloneqq \Gamma^{-1}\fN\) is irreducible for \(M_{\Id_\T}\). So let \(V \subeq \fM\) be a reducing subspace for \(M_{\Id_\T}\). Then \([M_{\Id_\T},P_V] = 0\) and therefore, since \(M_{\Id_\T}\) is unitary, we also get

\[[M_{\Id_\T}^*,P_V] = [M_{\Id_\T}^*,P_V^*] = -[M_{\Id_\T},P_V]^* = -0^* = 0.\]

This implies that

\[[M_{\Id_\T}^n,P_V] = [M_{\Id_\T^n},P_V] = 0 \quad \forall n \in \Z.\]

By prop:PolynomialsDense we have

\[\overline{\spann \{\Id_\T^n: n \in \Z\}} = L^\infty(\T,\C),\]

where the closure is taken with respect to the weak-\(*\)-topology on \(L^\infty(\T,\C)\). Therefore

\[[M_f,P_V] = 0 \quad \forall f \in L^\infty(\T,\C).\]

Using Cayley transform to switch the the upper half plane, setting \(\tilde V \coloneqq \Gamma V\) we get

\[[M_f,P_{\tilde V}] = 0 \quad \forall f \in L^\infty(\R,\C).\]

Since \(S_t = M_{e_t}\) for the functions \(e_t \in L^\infty(\R,\C)\) with \(e_t(x) \coloneqq e^{itx}\), we get

\[[S_t,P_{\tilde V}] = 0 \quad \forall t \in \R.\]

Therefore, for every \(t \in \R\), we get

\[\tilde V = P_{\tilde V} \tilde V= S_t S_{-t} P_{\tilde V} \tilde V = S_t P_{\tilde V} S_{-t} \tilde V \subeq S_t \tilde V = S_t \Gamma V \subeq S_t \Gamma \fM = S_t \fN,\]

\[\tilde V \subeq \bigcap_{t \in \R}S_t \fN = \{0\}\]

which yields \(\tilde V = \{0\}\) and therefore also

\[V = \Gamma^{-1} \tilde V = \{0\}.\]

This shows that no non-trivial subspace of \(\fM\) reduces \(U\) and therefore \(\fM\) is irreducible. Further, by prop:etDense, we have

\begin{align*}
M_{\Id_\T} \fM &\subeq H^\infty(\D,\C) \cdot \fM = H^\infty(\D,\C) \cdot \Gamma^{-1}\fN = \Gamma^{-1} \left(H^\infty(\C_+,\C) \cdot \fN\right)
\\&= \Gamma^{-1} \left(\overline{\bigcup_{t \geq 0} M_{e_t} \fN}\right) = \Gamma^{-1} \left(\overline{\bigcup_{t \geq 0} S_t \fN}\right) \subeq \Gamma^{-1} \fN = \fM.
\end{align*}

Therefore we can apply thm:Halmos and get a rigid function \(\tilde F \in L^\infty(\T,\mathrm{B}(\cK))\) such that

\[\fM = M_{\tilde F} H^2(\D,\cK).\]

Transforming to the upper half plane, for the rigid function

\[F \coloneqq \tilde F \circ h \in L^\infty(\R,\mathrm{B}(\cK)),\]

we get

\[\fN = \Gamma^{-1}\fM = \Gamma^{-1}M_{\tilde F} H^2(\D,\cK) = M_F \Gamma^{-1} H^2(\D,\cK) = M_F H^2(\C_+,\cK). \qedhere\]

Given a complex Hilbert space \(\cK\) and a closed subspace \(\cE_+\subeq L^2(\R,\cK)\) such that the triple \((L^2(\R,\cK),\cE_+,S)\) is a ROPG, by the Lax–Phillips Theorem (thm:LaxPhillipsComplex), there exists another complex Hilbert space \(\cK'\) such that \((L^2(\R,\cK),\cE_+,S)\) is equivalent to \((L^2(\R,\cK'),H^2(\C_+,\cK'),S)\). The following result tells us that in fact \(\cK' = \cK\):

Let \(\cK\) be a complex Hilbert space and \(\cE_+ \subeq L^2(\R,\cK)\) be a closed subspace such that \((L^2(\R,\cK),\cE_+,S)\) is a ROPG. Then \((L^2(\R,\cK),\cE_+,S)\) is equivalent to \((L^2(\R,\cK),H^2(\C_+,\cK),S)\).

By thm:HalmosUpper there exists a rigid function \(F \in L^\infty(\R,\mathrm{B}(\cK))\) such that

\[\cE_+ = M_F H^2(\C_+,\cK).\]

Now, we define

\[P: L^2(\R,\cK) \to L^2(\R,\cK), \quad (Pf)(x) = P_{F(x)\cK} f(x).\]

Then \(P\) is a projection with

\[PM_F = M_F \qquad \text{and} \qquad P \circ S_t = S_t \circ P, \quad t \in \R.\]

Now, for \(t \in \R\), we have

\[S_t \cE_+ = S_t M_F H^2(\C_+,\cK) = S_t P M_F H^2(\C_+,\cK) = P S_t M_F H^2(\C_+,\cK) \subeq P L^2(\R,\cK),\]

\[L^2(\R,\cK) = \overline{\bigcup_{t \in \R} S_t \cE_+} \subeq P L^2(\R,\cK).\]

This yields \(P = \textbf{1}\) and therefore \(P_x = \textbf{1}\) for almost every \(x \in \R\), which implies that \(F(x)\cK = \cK\) for almost every \(x \in \R\). Choosing such an element \(x_0 \in \R\), the operator \(F(x_0)\) then is a partial isometry with initial subspace \(\fU\) and final subspace \(\cK\) and therefore \(I \coloneqq F(x_0)^*\) is an isometry with initial subspace \(\cK\) and final subspace \(\fU\). We now define \(G \in L^\infty(\R,\mathrm{B}(\cK))\) by

\[G(x) \coloneqq F(x)I.\]

\[M_G H^2(\C_+,\cK) = M_F I H^2(\C_+,\cK) = M_F H^2(\C_+,\fU) = M_F H^2(\C_+,\cK) = \cE_+,\]

identifying \(H^2(\C_+,\fU)\) with the subspace

\[\{f \in H^2(\C_+,\cK): (\forall z \in \C_+) \,f(z) \in \fU\} \subeq H^2(\C_+,\cK).\]

Further, since \(I\) is an isometry with initial subspace \(\cK\) and final subspace \(\fU\) and for almost every \(x \in \R\) the operator \(F(x)\) is a partial isometry with initial subspace \(\fU\) and final subspace \(\cK\), for almost every \(x \in \R\) the operator \(G(x)\) is a partial isometry with initial subspace \(\cK\) and final subspace \(\cK\) and therefore \(G(x)\) is unitary. This implies that also \(M_G\) is unitary. We now set

\[\psi \coloneqq M_G^{-1} \in \U(L^2(\R,\cK)).\]

\[\psi\left(\cE_+\right) = H^2(\C_+,\cK)\]

\[\psi \circ S_t = S_t \circ \psi \quad \forall t \in \R. \qedhere\]

Let \(\cE\) be a Hilbert space and \(U\) be a unitary one-parameter group on \(\cE\). Further let \(\cE_+,\cE_+' \subeq \cE\) be two closed subspaces such that \((\cE,\cE_+,U)\) and \((\cE,\cE_+',U)\) are ROPGs. Then \((\cE,\cE_+,U)\) is equivalent to \((\cE,\cE_+',U)\).

We first consider the case of complex ROPGs. Then, by the Lax–Phillips Theorem (thm:LaxPhillipsComplex), there exists a complex Hilbert space \(\cK\) such that the ROPG \((\cE,\cE_+,U)\) is equivalent to \((L^2(\R,\cK),H^2(\C_+,\cK),S)\). Let \(\psi\) be the corresponding isomorphism of ROPGs. Then \((\cE,\cE_+',U)\), using the same isomorphism \(\psi\), is equivalent to \((L^2(\R,\cK),\psi \cE_+',S)\). By cor:StandardIndependentFromSubspace we know that \((L^2(\R,\cK),\psi \cE_+',S)\) is equivalent to \((L^2(\R,\cK),H^2(\C_+,\cK),S)\) and therefore also \((\cE,\cE_+,U)\) and \((\cE,\cE_+',U)\) are equivalent.

In the case of real ROPGs, we notice that, by prop:ROPGRealToComplex, \((\cE,\cE_+,U)\) and \((\cE,\cE_+',U)\) are equivalent, if and only if \((\cE_\C,(\cE_+)_\C,U_\C)\) and \((\cE_\C,(\cE_+')_\C,U_\C)\) are equivalent, which, as just seen, is indeed the case.

This theorem shows that, fixing a Hilbert space \(\cE\) and a unitary one-parameter group \(U\) on \(\cE\), there is — up to equivalence — only one possible choice of a subspace \(\cE_+\) such that \((\cE,\cE_+,U)\) is a ROPG.

§ ROPGS GENERATED BY PICK FUNCTIONS

In the last section we answered the question, given a ROPG \((\cE,\cE_+,U)\), what happens when one fixes the Hilbert space \(\cE\) and the unitary one-parameter group \(U\) and varies the subspace \(\cE_+\). In this chapter we want to answer the question, what happens, when one fixes the Hilbert space \(\cE\) and the subspace \(\cE_+\) and varies the unitary one-parameter group \(U\), so we will have two ROPGs of the form \((\cE,\cE_+,U)\) and \((\cE,\cE_+,U')\). In the complex case, by the Lax–Phillips Theorem, we then find a complex Hilbert space \(\cH\) such that \((\cE,\cE_+,U)\) is equivalent to \((L^2(\R,\cH),H^2(\C_+,\cH),S)\) via some isomorphism \(\psi\) of ROPGs. Using that same isomorphism \(\psi\), the ROPG \((\cE,\cE_+,U')\) is equivalent to \((L^2(\R,\cH),H^2(\C_+,\cH),\psi U'\psi^{-1})\). Therefore our question of interest is, for which unitary one-parameter groups \(U\) the triple \((L^2(\R,\cH),H^2(\C_+,\cH),U)\) is a complex ROPG and if so, how to calculate the corresponding multiplicity space.

§.§ Regular Pick functions

We will not give a complete characterization of all unitary one-parameter groups \(U\) such that the triple \((L^2(\R,\cH),H^2(\C_+,\cH),U)\) is a complex ROPG, but we will do so under two more assumptions. The first assumption is, that our multiplicity space \(\cH\) is finite dimensional. Our second assumption is, that the unitary one-parameter group \(U\) and our standard unitary one-parameter group \(S\) commute, i.e.

\[U_t S_s = S_s U_t \quad \forall s,t \in \R.\]

This last condition, by prop:SCommutant, is equivalent to \(U_t \in M \left(L^\infty(\R,B(\cH))\right)\) for every \(t \in \R\). Then the condition

\[U_t H^2(\C_+,\cH) \subeq H^2(\C_+,\cH) \quad \forall t \geq 0,\]

by prop:H2InclusionFunctions, implies that

\[U_t \in M\left(H^\infty(\C_+,B(\cH))\right) \quad \forall t \geq 0.\]

The unitarity of \(U_t\) then implies that

\[U_t \in M\left(\Inn(\C_+,B(\cH))\right) \quad \forall t \geq 0\]

(cf. def:Inner), which, by thm:OPGOfInnerFunctions, implies that \(U\) is infinitesimally generated by a Pick function, defined as follows:
(cf. def:Pick)
Let \(\cH\) be a finite dimensional complex Hilbert space.

* We call a function \({F \in \cO(\C_+,B(\cH))}\) an operator-valued Pick function if

\[\Im(F(z)) \coloneqq \frac 1{2i} (F(z)-F(z)^*) \geq 0, \quad \forall z \in \C_+.\]

* We write \(\mathrm{Pick}(\C_+,B(\cH))\) for the set of operator-valued Pick functions.

* We write \(\mathrm{Pick}(\C_+,B(\cH))_\R\) for the set of Pick functions \(F \in \mathrm{Pick}(\C_+,B(\cH))\), for which the limit

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

exists for almost every \(x \in \R\) and is self-adjoint. We say such a Pick function has self-adjoint boundary values.

* We set

\[\mathrm{Pick}(\C_+) \coloneqq \mathrm{Pick}(\C_+,B(\C)) \quad \text{and} \quad \mathrm{Pick}(\C_+)_\R \coloneqq \mathrm{Pick}(\C_+,B(\C))_\R\]

and in the latter case we say these functions have real boundary values.

Since we want to analyze ROPGs, we are interested in Pick functions, that are infinitesimal generators for ROPGs. Here we make the following definition:

Let \(\cH\) be a finite dimensional complex Hilbert space. We call a Pick function \(F \in \mathrm{Pick}(\C_+,B(\cH))_\R\) regular if

\[\bigcap_{t \in \R}e^{itF_*} \cdot H^2(\C_+,\cH) = \{0\} \quad \text{and} \quad \overline{\bigcup_{t \in \R}e^{itF_*} \cdot H^2(\C_+,\cH)} = L^2(\R,\cH).\]

We write \(F \in \mathrm{Pick}(\C_+,B(\cH))_\R^\mathrm{reg}\) for the set of all regular Pick functions.

Given this definition, a Pick function \(F \in \mathrm{Pick}(\C_+,B(\cH))_\R\) is regular, if and only if the triple \((L^2(\R,\cH),H^2(\C_+,\cH),U^F)\) with

\[\gls*{UF}_t \coloneqq M_{e^{itF_*}}, \quad t \in \R,\]

is a complex ROPG. Here we use that the condition

\[e^{itF_*} \cdot H^2(\C_+,\cH) \subeq H^2(\C_+,\cH) \quad \forall t \geq 0\]

is fulfilled for every Pick function \(F \in \mathrm{Pick}(\C_+,B(\cH))_\R\), since \(e^{itF} \in H^\infty(\C_+,B(\cH))\) for all \(t \geq 0\).

The goal of this section now is to give a classification of all regular Pick functions. For this we will use the fact, that the boundary values of a Pick function are a measurable function (cf. lem:BoundaryMeasurable) and therefore can be measurably diagonalized using the following results:
(cf. <cit.>)
Let \(X\) be a measurable space and \(n \in \N\). Further let \({F: X \to \mathrm{Mat}(n \times n)}\) be measurable such that \(F(x)\) is a positive definite matrix for every \(x \in X\). Then there exist measurable functions \(\Lambda,U: X \to \mathrm{Mat}(n \times n)\) such that \(\Lambda(x)\) is diagonal, \(U(x)\) is unitary and \(F(x) = U(x) \Lambda(x) U(x)^*\) for all \(x \in X\).

Let \(X\) be a measurable space and \(n \in \N\). Further let \({F: X \to \mathrm{Mat}(n \times n)}\) be measurable such that \(F(x)\) is a self-adjoint matrix for every \(x \in X\). Then there exist measurable functions \(\Lambda,U: X \to \mathrm{Mat}(n \times n)\) such that \(\Lambda(x)\) is diagonal, \(U(x)\) is unitary and \(F(x) = U(x) \Lambda(x) U(x)^*\) for all \(x \in X\).

By applying thm:measurableDiagonalizationPos to the function \(\exp \circ F\), we obtain measurable functions \(\tilde \Lambda, U\) such that \(\tilde \Lambda(x)\) is diagonal, \(U(x)\) is unitary and \((\exp \circ F)(x) = U(x) \tilde \Lambda(x) U(x)^*\) for all \(x \in X\). The statement then follows by choosing \(\Lambda \coloneqq \log \circ \tilde \Lambda\).

We now can formulate our first classfication theorem of regular Pick functions using the following definition:

For a Pick function \(F \in \mathrm{Pick}(\C_+,B(\cH))_\R\) and \(M \subeq \R\), we define

\[\gls*{DFM} \coloneqq \{y \in \R: F_*(y) \text{ exists and } M \cap \Spec(F_*(y)) \neq \eset\}.\]

Further, for \(x \in \R\), we set

\[D(F,x) \coloneqq D(F,\{x\}) = \{y \in \R: F_*(y) \text{ exists and } x \in \Spec(F_*(y))\}.\]

Let \(\cH\) be a finite dimensional complex Hilbert space and \(F \in \mathrm{Pick}(\C_+,B(\cH))_\R\). Then the following are equivalent:

* \(F\) is regular.

* \(\Spec(F(z)) \subeq \C_+\) 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 \(x \in \R\).

(a) \(\Rightarrow\) (c): Let \(F\) be regular and \(x \in \R\). Further let \(v \in \ker(x\textbf{1}-M_{F_*}) \cap H^2(\C_+,\cH)\). Then

\[\{0\} = \bigcap_{t \in \R}e^{itF_*}H^2(\C_+,\cH) \supeq \bigcap_{t \in \R}e^{itF_*}\C v = \bigcap_{t \in \R}e^{itx}\C v = \C v,\]

which implies \(v=0\).

(b) \(\Rightarrow\) (a): Assume that \(\Spec(F(z)) \subeq \C_+\) for all \(z \in \C_+\). Let \(h \in \bigcap_{t \in \R}e^{itF_*}H^2(\C_+,\cH)\). Now, for \(z \in \C_+\), let \(F(z)^*= D(z) + N(z)\) be the Jordan–Chevalley Decomposition of \(F(z)^*\) into a diagonalizable matrix \(D(z)\) and a nilpotent matrix \(N(z)\). Further, setting \(n \coloneqq \dim \cH\), we choose a basis \(v_1(z),\dots,v_n(z)\) of \(\cH\) of eigenvectors of \(D(z)\) and let \(\lambda_1(z),\dots,\lambda_n(z)\) be the corresponding eigenvalues, so

\[D(z)v_k(z) = \lambda_k(z)v_k(z)\]

for all \(k=1,\dots,n\). Then, for every \(t \in \R\) and \(k=1,\dots,n\), we have

\begin{align*}
\braket*{Q_z \cdot v_k(z)}{e^{itF_*}h} &= \braket*{v_k(z)}{e^{itF(z)}h(z)} = \braket*{e^{-itF(z)^*}v_k(z)}{h(z)}
\\&= \braket*{e^{-it\lambda_k(z)}\left(\sum_{j=0}^n \frac 1{j!}\left(-itN(z)\right)^j\right)v_k(z)}{h(z)}.
\end{align*}

with \(Q_z\) as in prop:H2RKHS.
This yields

\begin{align*}
\frac{e^{-it\overline{\lambda_k(z)}}}{t^n} \braket*{Q_z \cdot v_k(z)}{e^{itF_*}h} = \braket*{\left(\sum_{j=0}^n \frac 1{j!t^{n-j}}\left(-iN(z)\right)^j\right)v_k(z)}{h(z)}.
\end{align*}

We further have

\[\left|\braket*{Q_z \cdot v_k(z)}{e^{itF_*}h}\right| \leq \left\lVert Q_z\right\rVert \cdot \left\lVert v_k(z)\right\rVert \cdot \left\lVert h\right\rVert \quad \text{and} \quad \left|\frac{e^{-it\overline{\lambda_k(z)}}}{t^n}\right| = \frac{e^{-t\Im(\lambda_k(z))}}{t^n} \xrightarrow{t \to \infty} 0,\]

using that by assumption \(\lambda_k(z) \in \C_+\) and therefore \(\Im(\lambda_k(z)) > 0\). Then, for \(t \to \infty\), we get

\[0 = \braket*{\frac 1{n!}\left(-iN(z)\right)^n v_k(z)}{h(z)}.\]

\[\braket*{Q_z \cdot v_k(z)}{e^{itF_*}h} = \braket*{e^{-it\lambda_k(z)}\left(\sum_{j=0}^{n-1} \frac 1{j!}\left(-itN(z)\right)^j\right)v_k(z)}{h(z)}.\]

Now, repeating the same argument, one shows that

\[0 = \braket*{\frac 1{j!}\left(-iN(z)\right)^j v_k(z)}{h(z)}\]

for all \(j=0,\dots,n\). Especially, for \(j=0\), we get

\[0 = \braket*{v_k(z)}{h(z)}.\]

Since this is true for all \(k=1,\dots,n\) and the vectors \(v_1(z),\dots,v_n(z)\) form a basis of \(\cH\), we have \(h(z) = 0\) for every \(z \in \C_+\) and therefore \(h = 0\), so

\[\bigcap_{t \in \R}e^{itF_*}H^2(\C_+,\cH) = \{0\}.\]

On the other hand, using the same argument for the operator-valued Pick function \(F^\flat\) on the lower half plane \(\C_-\) defined by

\[F^\flat(z) \coloneqq -F(\overline{z})^*, \quad z \in \C_+,\]

one gets

\[\bigcap_{t \in \R} e^{it F^\flat_*}H^2(\C_-,\cH) = \{0\}.\]

Then, using that

\[F^\flat_* = -F^*_* = -F_* \quad \text{and} \quad H^2(\C_-,\cH) = H^2(\C_+,\cH)^\perp,\]

as seen in rem:ROPGAlternative, we get

\begin{align*}
\overline{\bigcup_{t \in \R}e^{itF_*}H^2(\C_+,\cH)} &= \left(\bigcap_{t \in \R} e^{it F^\flat_*}H^2(\C_-,\cH)\right)^\perp  = \{0\}^\perp = L^2(\R,\cH),
\end{align*}

so \(F\) is regular.

(c) \(\Rightarrow\) (b): Assume that \(\Spec(F(z)) \not\subeq \C_+\) for some \(z \in \C_+\). Since, by prop:SpecImPos(a), we have \(\Spec(F(z)) \subeq \overline{\C_+}\), there exists \(x \in \R\) and a \(v \in \cH \setminus \{0\}\) with \(F(z)v = xv\). Then, by prop:SpecImPos(b), we have \(\Im(F(z))v = 0\). We now define the Pick function

\[f: \C_+ \to \C, \quad w \mapsto \braket*{v}{F(w)v}.\]

\[\Im(f(z)) = \braket*{v}{\Im(F(z))v} = 0\]

and therefore, by lem:PickMaxModPrin, we get that \(f\) is constant. So

\[\braket*{v}{\Im(F(w))v} = \Im(f(w)) = \Im(f(z)) = \Im(x) = 0\]

for all \(w \in \C_+\). Since \(\Im(F(w)) \geq 0\), this implies that

\[0 = \Im(F(w))v = \frac 1{2i}(F(w)-F(w)^*)v\]

and therefore \(F(w)^*v = F(w)v\) for all \(w \in \C_+\).
We now define the function

\[g: \C_+ \to \C, \quad w \mapsto \left\lVert(F(w)-x)v\right\rVert^2.\]

\begin{align*}
g(w) &= \braket*{F(w)v}{F(w)v} - x \braket*{F(w)v}{v} - x \braket*{v}{F(w)v} + x^2 \braket*{v}{v}
\\&= \braket*{F(w)^*v}{F(w)v} - x \braket*{F(w)^*v}{v} - x \braket*{v}{F(w)v} + x^2 \left\lVert v\right\rVert^2
\\&= \braket*{v}{F(w)^2v} - 2x \braket*{v}{F(w)v} + x^2 \left\lVert v\right\rVert^2,
\end{align*}

the function \(g\) is holomorphic with \(\Im(g) \equiv 0\), which implies that \(g\) is constant. Therefore, for every \(w \in \C_+\), we have

\[\left\lVert(F(w)-x)v\right\rVert^2 = g(w) = g(z) = 0,\]

which implies

\[F(w)v = xv.\]

Taking limits, this also implies

\[F_*(y)v = xv \qquad \forall y \in \R.\]

Now, for every function \(h \in H^2(\C_+) \setminus \{0\}\), we have have \(h \otimes v \in H^2(\C_+,\cH) \setminus \{0\}\) with

\[(x\textbf{1}-M_{F_*})(h \otimes v) = 0,\]

so especially \(\ker(x\textbf{1}-M_{F_*}) \cap H^2(\C_+,\cH) \neq \{0\}\).

(d) \(\Rightarrow\) (c): Assume that there exists \(v \in \ker(x\textbf{1}-M_{F_*}) \cap H^2(\C_+,\cH)\) with \(v \neq 0\) for some \(x \in \R\). Since \(v \in H^2(\C_+,\cH)\), by thm:HardyAlmostEverywhereNonZero, we have \(v(y) \neq 0\) for almost every \(y \in \R\) and therefore

\[\ker(x\textbf{1}-F_*(y)) \supeq \C v(y) \neq \{0\}\]

for almost every \(y \in \R\). This implies that the Lebesgue measure of \(\R \setminus D(F,x)\) is zero.

(e) \(\Rightarrow\) (d): This follows immediately from the fact that \(\R\) has non-zero Lebesgue measure.

(c) \(\Rightarrow\) (e): Assume that \(D(F,x)\) is no Lebesgue zero-set for some \(x \in \R\). Then there is a \(R>0\) such that \(\lambda_1(D(F,x) \cap [-R,R])>0\), denoting by \(\lambda_1\) the Lebesgue measure on \(\R\). Setting \(n \coloneqq \dim \cH\), by cor:measurableDiagonalization there exists a measurable function \(U: \R \to \mathrm{Mat}(n \times n)\) such that \(U(x)\) is unitary for all \(x \in \R\) and functions \(d_1,\dots,d_n: \R \to \R\) such that, after the identification \(\cH \cong \C^n\), we have

\[F_*(y) = U(y) \diag(d_1(y),\dots,d_n(y))U(y)^*\]

for almost every \(y \in \R\). For \(k = 1,\dots,n\), we define the measurable sets

\[M_k \coloneqq d_k^{-1}(x) \setminus \bigcup_{j=1}^{k-1}d_j^{-1}(x).\]

Denoting by \(\{e_1,\dots,e_n\}\) the canonical ONB of \(\cH \cong \C^n\), we now consider the measurable function

\[g \coloneqq \sum_{k=1}^n \chi_{M_k \cap [-R,R]}Ue_k.\]

One has

\[\left\lVert g\right\rVert^2 = \sum_{k=1}^n \lambda_1(M_k \cap [-R,R]) = \lambda_1(D(F,x) \cap [-R,R]) \in ]0,2R],\]

so \(g \in L^2(\R,\cH) \setminus \{0\}\).
Further, for almost every \(y \in \R\), we have

\begin{align*}
F_*(y)g(y) &= \sum_{k=1}^n \chi_{M_k \cap [-R,R]}(y)F_*(y)U(y)e_k = \sum_{k=1}^n \chi_{M_k \cap [-R,R]}(y)d_k(y)U(y)e_k
\\&= \sum_{k=1}^n \chi_{M_k \cap [-R,R]}(y)xU(y)e_k = xg(y),
\end{align*}

so \(g \in \ker(x\textbf{1}-M_{F_*})\). Since \(g\big|_{\R \setminus [-R,R]} \equiv 0\), we know that \(g \notin H^2(\C_-,\cH)\) by thm:HardyAlmostEverywhereNonZero. Denoting by \(P_\pm\) the projections onto \(H^2(\C_\pm,\cH)\) and setting \(g_\pm \coloneqq P_\pm g\), we therefore have \(g_+ \neq 0\). Now, for every \(t \in \R\), we have

\[e^{itx}g_+ + e^{itx}g_- = e^{itx}g = e^{itF_*}g = e^{itF_*}g_+ + e^{itF_*}g_-.\]

For \(t \geq 0\) we have \(e^{itF_*}g_+ \in H^2(\C_+,\cH)\), so applying \(P_\pm\) to both sides, we get

\[e^{itx}g_+ = e^{itF_*}g_+ + P_+ e^{itF_*}g_- \quad \text{and} \quad e^{itx}g_- = P_- e^{itF_*}g_-.\]

This implies

\[\left\lVert P_+ e^{itF_*}g_-\right\rVert^2 = \left\lVert e^{itF_*}g_-\right\rVert^2 - \left\lVert P_- e^{itF_*}g_-\right\rVert^2 = \left\lVert g_-\right\rVert^2 - \left\lVert e^{itx}g_-\right\rVert^2 = 0,\]

so \(P_+ e^{itF_*}g_- = 0\) and therefore

\[e^{itx}g_+ = e^{itF_*}g_+ \quad \text{and} \quad e^{itx}g_- = e^{itF_*}g_-\]

for all \(t \geq 0\). This implies that

\[g_+,g_- \in \ker(x\textbf{1}-M_{F_*}),\]

so since \(g_+ \neq 0\), we especially get

\[\ker(x\textbf{1}-M_{F_*}) \cap H^2(\C_+,\cH) \neq \{0\}. \qedhere\]

We now will see some consequences of this theorem:

A function \(f \in \mathrm{Pick}(\C_+)_\R\) is regular, if and only if \(f\) is non-constant.

If \(f \in \mathrm{Pick}(\C_+)_\R\) is constant, than, since \(f\) has real boundary values, there is a \(\lambda \in \R\) with \(f \equiv \lambda\), so \(\Im f(z) = 0\) for every \(z \in \C_+\). If on the other hand \(f \in \mathrm{Pick}(\C_+)_\R\) is non-constant, then, by lem:PickMaxModPrin, we have \(\Im f(z) > 0\) for every \(z \in \C_+\). Therefore \(f\) is non-constant, if and only if

\[\Spec(f(z)) = \{f(z)\} \subeq \C_+\]

for every \(z \in \C_+\) which, by thm:SpecNonDeg, is equivalent to the function \(f\) being regular.

Since, by thm:SpecNonDeg, for \(f \in \mathrm{Pick}(\C_+)_\R^{\mathrm{reg}}\) and \(F \in \mathrm{Pick}(\C_+,B(\cH))_\R^{\mathrm{reg}}\), for \(z \in \C_+\), one has

\[f(z) \in \C_+ \quad \text{and} \quad \Spec(f(z)) \subeq \C_+,\]

the operators

\[F(f(z)) \in B(\cH) \quad \text{and} \quad f(F(z)) \in B(\cH)\]

are defined. Therefore, the maps

\[\circ: \mathrm{Pick}(\C_+,B(\cH))_\R^{\mathrm{reg}} \times \mathrm{Pick}(\C_+)_\R^{\mathrm{reg}} \to \mathrm{Pick}(\C_+,B(\cH)), \quad (F \circ f)(z) \coloneqq F(f(z))\]

\[\circ: \mathrm{Pick}(\C_+)_\R^{\mathrm{reg}} \times \mathrm{Pick}(\C_+,B(\cH))_\R^{\mathrm{reg}} \to \mathrm{Pick}(\C_+,B(\cH)), \quad (f \circ F)(z) \coloneqq f(F(z))\]

are also defined. The following theorem shows that their image in fact is again \(\mathrm{Pick}(\C_+,B(\cH))_\R^{\mathrm{reg}}\):

Let \(\cH\) be a finite dimensional complex Hilbert space. Further, let \(f,g \in \mathrm{Pick}(\C_+)_\R\) and \(F \in \mathrm{Pick}(\C_+,B(\cH))_\R\) all be regular. Then \(f \circ F \circ g\) is also regular.

By thm:SpecNonDeg, for \(z \in \C_+\), we have

\[\{f(z)\} = \Spec(f(z)) \subeq \C_+, \quad \{g(z)\} = \Spec(g(z)) \subeq \C_+ \quad \text{and} \quad \Spec(F(z)) \subeq \C_+.\]

Therefore, for \(z \in \C_+\), we get

\[\Spec((f \circ F \circ g)(z)) = f(\Spec(F(g(z)))) \subeq f(\C_+) \subeq \C_+,\]

which by thm:SpecNonDeg implies that \(f \circ F \circ g\) is regular.

With help of this corollary, we can prove a theorem with a generalization of condition (e) in thm:SpecNonDeg:

Let \(\cH\) be a finite dimensional complex Hilbert space and \(F \in \mathrm{Pick}(\C_+,B(\cH))_\R\). Then the following are equivalent:

* \(F\) is regular.

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

(b) \(\Rightarrow\) (a): Follows immediately from thm:SpecNonDeg, using the fact that \(\{x\}\) is a Lebesgue zero-set for every \(x \in \R\).

(a) \(\Rightarrow\) (b): We assume that there is a Lebesgue zero-set \(M \subeq \R\) such that \(D(F,M)\) has positive Lebesgue measure. By cor:measurableDiagonalization there exists a measurable function \(U: \R \to \mathrm{Mat}(n \times n)\) such that \(U(x)\) is unitary for all \(x \in \R\) and measurable functions \(d_1,\dots,d_n: \R \to \R \cup \{\infty\}\) such that, after the identification \(\cH \cong \C^n\), we have

\[F_*(y) = U(y) \diag(d_1(y),\dots,d_n(y))U(y)^*\]

for every \(y \in \R\) for which \(F_*(y)\) exists and \(d_1(y),\dots,d_n(y) = \infty\) for every \(y \in \R\) for which \(F_*(y)\) doesn't exist. Then

\[D(F,M) = \bigcup_{k=1}^n d_k^{-1}(M).\]

So, by assumption, there exists \(K \in \{1,\dots,n\}\) such that \(d_K^{-1}(M)\) has positive Lebesgue measure. We now choose \(R>0\) such that

\[\lambda_1\left(d_K^{-1}(M) \cap [-R,R]\right)>0.\]

\[\mu(E) \coloneqq \lambda_1\left(d_K^{-1}(M \cap E) \cap [-R,R]\right)\]

defines a measure \(\mu\) on \(\R\) with

\[\mu(\R) = \lambda_1\left(d_K^{-1}(M) \cap [-R,R]\right),\]

which implies

\[0 < \mu(\R) \leq 2R < \infty,\]

so \(\mu\) is non-zero and finite. Further, since \(M\) is a Lebesgue zero-set and \({\mu(\R \setminus M) = 0}\), it is also Lebesgue-singular. Therefore, by thm:GePick,

\[F_\mu(z) \coloneqq \int_\R \frac 1{\lambda-z} - \frac \lambda{1+\lambda^2} \,d\mu(\lambda)\]

defines a Pick function with real boundary-values. Further, by thm:GePick, setting

\[S_\mu \coloneqq \left\{x \in \R: \lim_{\epsilon \downarrow 0} \Im(F_\mu(x+i\epsilon)) = +\infty\right\},\]

we have \(\mu(E) = \mu(E \cap S_\mu)\) for every measurable subset \(E \subeq \R\). In particular this implies that

\[0 \neq \mu(\R) = \mu(S_\mu) = \lambda_1\left(d_K^{-1}(M \cap S_\mu) \cap [-R,R]\right).\]

Further, defining the Pick function

\[\eta: \C_+ \to \C_+, \quad z \mapsto -\frac 1z,\]

we get

\[(\eta \circ F_\mu)_*(x) = 0\]

for every \(x \in S_\mu\). For \(x \in d_K^{-1}(M \cap S_\mu) \cap [-R,R]\), by definition, we have that

\[\eset \neq (M \cap S_\mu) \cap \Spec(F_*(x)),\]

\[\eset \neq (\eta \circ F_\mu)_*(M \cap S_\mu) \cap \Spec((\eta \circ F_\mu \circ F)_*(x)) = \{0\} \cap \Spec((\eta \circ F_\mu \circ F)_*(x))\]

and therefore

\[0 \in \Spec((\eta \circ F_\mu \circ F)_*(x)).\]

We then obtain

\[D(\eta \circ F_\mu \circ F,0) \supeq d_K^{-1}(M \cap S_\mu) \cap [-R,R]\]

and therefore \(D(\eta \circ F_\mu \circ F,0)\) is no Lebesgue zero-set, so, by thm:SpecNonDeg, we have that \(\eta \circ F_\mu \circ F\) is non-regular. Since \(\eta\) and \(F_\mu\) are non-constant and therefore, by cor:ConstNonDeg, regular, we get, by cor:CompNonDeg, that \(F\) is non-regular.

After now having some characterizations for regular Pick functions, i.e. the Pick functions \(F \in \mathrm{Pick}(\C_+,B(\cH))_\R\) such that the triple \((L^2(\R,\cH),H^2(\C_+,\cH),U^F)\) is a complex ROPG, we now want to show how to use these characterizations for real ROPGs. This question is answered in the following theorem:

Let \(\cH\) be a real Hilbert space and \(F \in \mathrm{Pick}(\C_+,B(\cH_\C))_\R\). Then \(U^F\) defines a unitary one-parameter group on \(L^2(\R,\cH_\C)^\sharp\), if and only if \(F = F^\sharp\) with

\[F^\sharp(z) \coloneqq -\cC_\cH F(-\overline{z})\cC_\cH, \quad z \in \C_+.\]

In this case the triple \((L^2(\R,\cH_\C)^\sharp,H^2(\C_+,\cH_\C)^\sharp,U^F)\) is a real ROPG with multiplicity space \(\cK\), if and only if the triple \((L^2(\R,\cH),H^2(\C_+,\cH_\C),U^F)\) is a complex ROPG with multiplicity space \(\cK_\C\).

One has

\[e^{itF_*} \cdot L^2(\R,\cH_\C)^\sharp \subeq L^2(\R,\cH_\C)^\sharp \quad \forall t \in \R,\]

if and only if

\begin{align*}
\left(e^{itF_*} \cdot f\right)(x) &= \left(e^{itF_*} \cdot f\right)^\sharp(x) = \cC_\cH e^{itF_*(-x)} f(-x)
\\&= e^{-it\cC_\cH F_*(-x)\cC_\cH} \cC_\cH f(-x) = e^{it F^\sharp_*(x)} f^\sharp(x) = e^{it F^\sharp_*(x)} f(x)
\end{align*}

for all \(t \in \R\), \(f \in L^2(\R,\cH_\C)^\sharp\) and almost all \(x \in \R\). This is the case, if and only if

\[F_*(x) = F^\sharp_*(x)\]

for almost all \(x \in \R\), which is equivalent to \(F = F^\sharp\). The second statement follows by prop:ROPGRealToComplex.

§.§ The multiplicity free case

In the last section, given a finite dimensional complex Hilbert space \(\cH\), we have seen for which Pick functions \({F \in \mathrm{Pick}(\C_+,B(\cH))_\R}\) the triple \((L^2(\R,\cH),H^2(\C_+,\cH),U^F)\) is a complex ROPG and likewise, given a finite dimensional real Hilbert space \(\cH\), for which \(F \in \mathrm{Pick}(\C_+,B(\cH_\C))_\R\) the triple \((L^2(\R,\cH_\C)^\sharp,H^2(\C_+,\cH_\C)^\sharp,U^F)\) is a real ROPG.

The goal of the following sections will be, given a regular Pick function \({F \in \mathrm{Pick}(\C_+,B(\cH))_\R}\) or \(F \in \mathrm{Pick}(\C_+,B(\cH_\C))_\R\) respectively, to calculate the multiplicty space for the complex ROPG \((L^2(\R,\cH),H^2(\C_+,\cH),U^F)\) or the real ROPG \((L^2(\R,\cH_\C)^\sharp,H^2(\C_+,\cH_\C)^\sharp,U^F)\) respectively.

We will later provide an answer in the general case of any finite dimensional Hilbert space \(\cH\). In this section, we will discuss the multiplicity free case \(\cH = \C\). Although not generalizable to a case with multiplicities, this proof gives a good idea of the behavior of Pick functions and how they produce multiplicities.

By thm:GePick, every function \(F \in \mathrm{Pick}(\C_+)_\R\) is of the form

\[F(z) = C + Dz + \int_\R \frac 1{\lambda - z} - \frac{\lambda}{1+\lambda^2} \,d\mu(\lambda)\]

with \(C \in \R\), \(D \geq 0\) and some singular Borel measure \(\mu\) on \(\R\) supported on the set

\[S_\mu \coloneqq \left\{x \in \R: \lim_{\epsilon \downarrow 0} \Im(F(x+i\epsilon)) = +\infty\right\}.\]

We set

\[S_F \coloneqq \begin{cases} S_\mu & \text{if }D = 0 \\ S_\mu \cup \{\infty\} & \text{if }D > 0\end{cases}\]

(cf. rem:GePick). With this definition we can formulate our theorem:

Let \(F \in \mathrm{Pick}(\C_+)_\R\) such that \(S_F\) is a non-empty closed subset of \(\overline{\R}\). Then \((L^2(\R,\C),H^2(\C_+),U^F)\) is a complex ROPG and its multiplicity space \(\cK\) is given by

\[\cK \coloneqq \begin{cases} \C^n & \text{if }|S_F| = n \in \N \\ \ell^2(\N,\C) & \text{if }|S_F| = \infty.\end{cases}\]

Since \(S_F\) is non-empty, the function \(F\) is non-constant and therefore, by cor:ConstNonDeg, the triple \((L^2(\R,\C),H^2(\C_+),U^F)\) is a complex ROPG. We now calculate its multiplicity space.

By lem:countableUnion there is a countable family \(\left(a_j,b_j\right)_{j \in J}\) in \(\overline{\R} \times \overline{\R}\) such that

\[\overline{\R} \setminus S_F = \bigsqcup_{j \in J} \left(a_j,b_j\right).\]

One has \(S_F = \{a_j: j \in J\}\) and therefore \(|S_F| = |J|\). Further, by prop:PickBoundaryDiffeo, for every \(j \in J\), the map

\[F_j \coloneqq F_*\big|_{\left(a_j,b_j\right)}: \left(a_j,b_j\right) \to \R\]

is monotonically increasing, bijective and continuously differentiable. Therefore, for every function \(g \in L^1(\R,\C) \cong L^1\left(\overline{\R},\C\right)\), one has

\begin{align*}
\int_\R g(x) \,dx &= \int_{\overline{\R} \setminus S_F} g(x) \,dx = \sum_{j \in J}\int_{(a_j,b_j)}g(x)\,dx = \sum_{j \in J}\int_\R g\left(F_j^{-1}(y)\right) \left(F_j^{-1}\right)'(y)\,dy.
\end{align*}

This implies that the map

\[V: L^2(\R,\C) \to \ell^2(J,L^2(\R,\C)), \quad f \mapsto \left(\sqrt{\left(F_j^{-1}\right)'} \cdot \left(f \circ F_j^{-1}\right)\right)_{j \in J}\]

is a unitary isomorphism of Hilbert spaces. Further, by the Fubini-Tonelli Theorem, the map

\[T: \ell^2(J,L^2(\R,\C)) \to L^2(\R,\ell^2(J,\C)), \quad T\left(\left(f_j\right)_{j\in J}\right)(x) = \left(f_j(x)\right)_{j \in J},\]

is a unitary isomorphism of Hilbert spaces. Then, for \(f \in L^2(\R,\C)\) and \(t \in \R\), one has

\begin{align*}
(T \circ V)(U^F_t f)(x) &= (T \circ V)\left(e^{itF_*} f\right)(x) = T\left(\left(\sqrt{\left(F_j^{-1}\right)'} \cdot e^{it\left(F_* \circ F_j^{-1}\right)}\left(f \circ F_j^{-1}\right)\right)_{j \in J}\right)(x)
\\&=\left(\sqrt{\left(F_j^{-1}\right)'(x)} \cdot e^{it\left(F_* \circ F_j^{-1}\right)(x)}\left(f \circ F_j^{-1}\right)(x)\right)_{j \in J}
\\&=\left(\sqrt{\left(F_j^{-1}\right)'(x)} \cdot e^{itx}\left(f \circ F_j^{-1}\right)(x)\right)_{j \in J}
\\&=e^{itx} \cdot \left(\sqrt{\left(F_j^{-1}\right)'(x)} \cdot \left(f \circ F_j^{-1}\right)(x)\right)_{j \in J}
\\&= e^{itx} \cdot \left(T \circ V\right)(f)(x) = \left(S_t \circ T \circ V\right)(f)(x),
\end{align*}

\[(T \circ V) \circ U^F_t = S_t \circ (T \circ V) \quad \forall t \in \R.\]

Now set

\[\cE_+ \coloneqq \left(T \circ V\right)\left(H^2(\C_+)\right) \subeq L^2(\R,\ell^2(J,\C)).\]

Then, for every \(t \in \R\), one has

\begin{align*}
S_t \cE_+ &= \left(S_t \circ \left(T \circ V\right)\right)\left(H^2(\C_+)\right) = \left(\left(T \circ V\right) \circ U^F_t\right)\left(H^2(\C_+)\right),
\end{align*}

so, using that \((L^2(\R,\C),H^2(\C_+),U^F)\) is a complex ROPG, we get

\[S_t \cE_+ = \left(T \circ V\right)\left(U^F_t H^2(\C_+)\right) \subeq \left(T \circ V\right)\left(H^2(\C_+)\right) = \cE_+ \quad \forall t \geq 0\]

\[\bigcap_{t \in \R} S_t \cE_+ = \left(T \circ V\right)\left(\bigcap_{t \in \R} U^F_t H^2(\C_+)\right) = \left(T \circ V\right)\left(\{0\}\right) = \{0\}\]

\[\overline{\bigcup_{t \in \R} S_t \cE_+} = \left(T \circ V\right)\left(\overline{\bigcup_{t \in \R} U^F_t H^2(\C_+)}\right) = \left(T \circ V\right)\left(L^2(\R,\C)\right) = L^2(\R,\ell^2(J,\C)).\]

This implies that the triple \((L^2(\R,\ell^2(J,\C)),\cE_+,S)\) is a complex ROPG, which is equivalent to \((L^2(\R,\C),H^2(\C_+),U^F)\) via the isomorphism \(T \circ V\). Further, by thm:IndependentFromSubspace the complex ROPG \((L^2(\R,\ell^2(J,\C)),\cE_+,S)\) is equivalent to \((L^2(\R,\ell^2(J,\C)),H^2(\C_+,\ell^2(J,\C)),S)\). So, setting \({\cK \coloneqq \ell^2(J,\C)}\), we have that \((L^2(\R,\C),H^2(\C_+),U^F)\) is equivalent to \((L^2(\R,\cK),H^2(\C_+,\cK),S)\) and therefore \(\cK = \ell^2(J,\C)\) is the multiplicity space of \((L^2(\R,\C),H^2(\C_+),U^F)\). The statement then follows since

\[\ell^2(J,\C) \cong \C^{|J|} = \C^{|S_F|}\]

if \(|J| = |S_F| \in \N\) and

\[\ell^2(J,\C) \cong \ell^2(\N,\C)\]

if \(|J| = |S_F| = \infty\).

One has \(|S_F| \in \N\), if and only if the measure in the Herglotz representation of \(F\) is supported on a finite set, i.e. if there are \(\lambda_1,\dots,\lambda_n \in \R\) and \(a_1,\dots,a_n \in \R_+\) such that

\[F(z) = C + Dz + \sum_{k=1}^n \frac{a_k}{\lambda_k-z}, \quad z \in \C_+,\]

for some \(C \in \R\) and \(D \in \R_{\geq 0}\). In this case, one has

\[|S_F| = \begin{cases} n & \text{if } D= 0 \\ n+1 & \text{if } D > 0.\end{cases}\]

We will now also provide an analogue to thm:PickMultiplicity in the real case:

Let \(F \in \mathrm{Pick}(\C_+)_\R\) such that \(S_F\) is a non-empty closed subset of \(\overline{\R}\). Then \((L^2(\R,\C)^\sharp,H^2(\C_+)^\sharp,U^F)\) is a real ROPG, if and only if

\[F(z) = - \overline{F(-\overline{z})} \quad \forall z \in \C_+\]

and in this case its multiplicity space \(\cK\) is given by

\[\cK \coloneqq \begin{cases} \R^n & \text{if }|S_F| = n \in \N \\ \ell^2(\N,\R) & \text{if }|S_F| = \infty.\end{cases}\]

This follows immediately by thm:RealToComplexPick and thm:PickMultiplicity.

The condition \(F(z) = - \overline{F(-\overline{z})}\), for a Pick function

\[F(z) = C + Dz + \int_\R \frac 1{\lambda-z}-\frac \lambda{1+\lambda^2} \,d\mu(\lambda)\]

is equivalent to the measure \(\mu\) being symmetric, i.e. \(\mu(A) = \mu(-A)\) for all measurable subsets \(A \subeq \R\). In this case, the Pick function can be written as

\begin{align*}
F(z) &= C + Dz - \frac{\mu(\{0\})}z + \int_{\R_+} \frac 1{\lambda-z}+\frac 1{-\lambda-z} \,d\mu(\lambda)
\\&= C + Dz - \frac{\mu(\{0\})}z + \int_{\R_+} \frac {2z}{\lambda^2-z^2} \,d\mu(\lambda), \qquad \qquad z \in \C_+.
\end{align*}

§.§ The general case

After considering the one-dimensional case, in this section, given any finite dimensional complex Hilbert space \(\cH\) and a regular Pick function \({F \in \mathrm{Pick}(\C_+,B(\cH))_\R}\), we want to calculate the multiplicity space for the complex ROPG \((L^2(\R,\cH),H^2(\C_+,\cH),U^F)\), as well as provide an analogue in the real case. We will do this by using thm:kernelAdjointGeneral, in which appears the operator

\[p\left(\partial U^F\right)p^* = p M_{F_*} p^* = M_F\]

with \(p \coloneqq p_{H^2(\C_+,\cH)}\).
Therefore, given a Pick function \({F \in \mathrm{Pick}(\C_+,B(\cH))_\R}\) and \(\lambda \in \C_+\), we have to better understand the kernel \(\ker\left(\overline{\lambda}\textbf{1}-M_F^*\right) \subeq H^2(\C_+,\cH)\). We will see that this kernel has a property that we will call \(M_{H^\infty}^*\)-stability. Therefore, the following subsection will be dedicated to the theory of \(M_{H^\infty}^*\)-stable subspaces.

§.§.§ \(M_{H^\infty}^*\)-stable subspaces and finite Blaschke–Potapov products

Throughout this subsection, by \(\perp\) we will denote the orthogonal complement in the Hilbert space \(L^2(\R,\cH)\) and by \(\gls*{ZZZO+}\) the orthogonal complement in the Hilbert space \(H^2(\C_+,\cH)\).

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

\[\gls*{P+} \coloneqq P_{H^2(\C_+,\cH)} \in B(L^2(\R,\cH)).\]

A subspace \(V \subeq H^2(\C_+,\cH)\) is called \(M_{H^\infty}^*\)-stable, if

\[P_+ M_\phi^* V \subeq V\]

for every \(\phi \in H^\infty(\C_+)\).

It turns out that \(M_{H^\infty}^*\)-stable subspaces are closely related to SROPGs:

Let \(\cH\) be a complex Hilbert space. A subspace \(V \subeq H^2(\C_+,\cH)\) is \(M_{H^\infty}^*\)-stable, if and only if \((L^2(\R,\cH),V^{\perp_+},S)\) is a SROPG.

We first assume that \(V\) is a \(M_{H^\infty}^*\)-stable subspace, so especially \(P_+S_t^* V \subeq V\) for every \(t \geq 0\) since \(S_t = M_{e_t}\) for the functions \(e_t \in L^\infty(\R,\C)\) with

\[\gls*{et}(x) \coloneqq e^{itx}.\]

Then, for \(t \geq 0\), \(f \in V^{\perp_+}\) and \(g \in V\), one has

\[\braket*{S_t f}{g} = \braket*{f}{S_t^* g} = \braket*{P_+ f}{S_t^* g} = \braket*{f}{P_+ S_t^* g} = 0,\]

using that \(P_+ S_t^* g \in V\). This shows that \(S_t V^{\perp_+} \subeq V^{\perp_+}\) for \(t \geq 0\). Since also

\[\bigcap_{t \in \R} S_t V^{\perp_+} \subeq \bigcap_{t \in \R} S_t H^2(\C_+,\cH) = \{0\},\]

we get that \((L^2(\R,\cH),V^{\perp_+},S)\) is a SROPG.

Now, conversely assume that \((L^2(\R,\cH),V^{\perp_+},S)\) is a SROPG. Then, for \(t \geq 0\), \(f \in V^{\perp_+}\) and \(g \in V\), one has

\[\braket*{f}{P_+ S_t^* g} = \braket*{S_t f}{g} = 0,\]

using that \(S_t f \in V^{\perp_+}\). This shows that

\[P_+ M_{e_t}^* V = P_+ S_t^* V \subeq V \quad t \geq 0,\]

which, by prop:etDense implies the statement.

Let \(\cH\) be a complex Hilbert space. A subspace \(V \subeq H^2(\C_+,\cH)\) is \(M_{H^\infty}^*\)-stable, if and only if it is of the form

\[V = \left(M_\phi H^2(\C_+,\cH)\right)^{\perp_+}\]

for some function \(\phi \in H^\infty(\C_+,B(\cH))\).

Let \(V \subeq H^2(\C_+,\cH)\) be \(M_{H^\infty}^*\)-stable. Then, by prop:stableCharakterizationSROPG and thm:HalmosUpper there exists a rigid function \(\phi \in L^\infty(\R,\mathrm{B}(\cH))\) such that

\[M_\phi H^2(\C_+,\cH) = V^{\perp_+} \subeq H^2(\C_+,\cH).\]

The last inclusion in particular implies \(\phi \in H^\infty(\C_+,\mathrm{B}(\cH))\) by prop:H2InclusionFunctions.

Conversely, for every \(\phi \in H^\infty(\C_+,\mathrm{B}(\cH))\), one has

\[S_t \left(M_\phi H^2(\C_+,\cH)\right) = M_\phi \left(S_t H^2(\C_+,\cH)\right) \subeq M_\phi H^2(\C_+,\cH) \quad \forall t \geq 0\]

\[\bigcap_{t \in \R} S_t \left(M_\phi H^2(\C_+,\cH)\right) \subeq \bigcap_{t \in \R} S_t H^2(\C_+,\cH) = \{0\}.\]

Therefore \((L^2(\R,\cH),M_\phi H^2(\C_+,\cH),S)\) is a SROPG, which, by prop:stableCharakterizationSROPG, is equivalent to \(\left(M_\phi H^2(\C_+,\cH)\right)^{\perp_+}\) being a \(M_{H^\infty}^*\)-stable subspace.

When working with \(M_{H^\infty}^*\)-stable subspaces, the following result is often usefull:

Let \(\cH\) be a complex Hilbert space and \(\phi,\psi \in H^\infty(\C_+,B(\cH))\). Then

\[P_+ M_\phi^* P_+ M_\psi^* = P_+ M_{\psi \cdot \phi}^*.\]

We have

\[P_+M_\phi^*(\textbf{1}-P_+)L^2(\R,\cH) = P_+M_\phi^*H^2(\C_-,\cH) \subeq P_+H^2(\C_-,\cH) = \{0\},\]

so \(P_+M_\phi^*(\textbf{1}-P_+) = 0\) and therefore

\[P_+M_\phi^*P_+ = P_+M_\phi^*,\]

which implies

\[P_+ M_\phi^* P_+ M_\psi^* = P_+ M_\phi^* M_\psi^* = P_+ \left(M_\psi M_\phi\right)^* = P_+ M_{\psi \cdot \phi}^*. \qedhere\]

Let \(\cH\) be a complex Hilbert space and \(V \subeq H^2(\C_+,\cH)\) be a finite dimensional \(M_{H^\infty}^*\)-stable subspace. Then, if \(V \neq \{0\}\), there exist \(\omega \in \C_+\) and \(v \in \cH \setminus \{0\}\) such that \(Q_\omega \cdot v \in V\).

Since \(V\) is \(M_{H^\infty}^*\)-stable, we have

\[P_+S_t^*V \subeq V \quad \forall t \geq 0.\]

Further, for \(s,t \geq 0\) and \(f \in V\), by lem:ProjectionEquation,
one has

\[\left(P_+S_s^*\right)\left(P_+S_t^*\right) = P_+S_{(s+t)}^*.\]

Since \(\left\lVert P_+S_t^*\right\rVert \leq 1\), this implies that

\[M_t \coloneqq (P_+S_t^*) \big|_V^V\]

defines a one-parameter semigroup of contractions. Therefore, by the Lumer–Phillips Theorem (cf. <cit.>), there exists an operator \(A \in B(V)\) such that

\[M_t = e^{tA} \quad \forall t \geq 0.\]

As an operator on a finite dimensional vector space, the operator \(A\) has at least one eigenvalue with a corresponding eigenvector, so there exist \(\mu \in \C\) and \(f \in V \setminus \{0\}\) such that \(A f = \mu f\). This implies that

\[e^{t\mu} f = e^{tA} f = M_t f \quad \forall t \geq 0.\]

Applying Fourier transform, for all \(t \geq 0\), we get

\[e^{t\mu} (\cF f)(p) = (\cF M_t f)(p) = (\cF f)(p+t)\]

for almost every \(p \in \R_+\). This implies that there exists a \(\tilde v \in \cH \setminus \{0\}\) such that

\[(\cF f)(p) = \begin{cases} e^{p\mu} \tilde v & \text{for } p > 0 \\ 0 & \text{for } p \leq 0,\end{cases}\]

using that \(f \in V \subeq H^2(\C_+,\cH)\), which yields that \(\cF f\) is supported on \(\R_+\). Then the fact that \(\cF f \in L^2(\R,\cH)\) implies \(\Re \mu < 0\). Setting \(\omega = -i \overline{\mu}\) we therefore have \(\omega \in \C_+\). Also we set \(v \coloneqq \sqrt{2\pi} \cdot \tilde v \neq 0\). Then, using that \(f \in V \subeq H^2(\C_+,\cH)\), for \(z \in \C_+\), inverse Fourier transformation gives us

\begin{align*}
f(z) &= \frac 1{\sqrt{2\pi}}\int_0^\infty e^{ipz}e^{p\mu} \,dp \cdot \tilde v = \frac{1}{2\pi}\int_0^\infty e^{ip(z-\overline{\omega})} \,dp \cdot v = \frac i{2\pi} \cdot \frac 1{z-\overline{\omega}} \cdot v = Q_\omega(z) \cdot v. \qedhere
\end{align*}

We will use this lemma to link \(M_{H^\infty}^*\)-stable subspaces to so called Blaschke–Potapov products, which are defined as follows:
(cf. [4])
Let \(\cH\) be a complex Hilbert space.

* For \(\omega \in \C_+\) we define \(\phi_\omega \in \Inn(\C_+)\) (cf. def:Inner) by

\[\phi_\omega(z) \coloneqq \frac{z-\omega}{z-\overline{\omega}}.\]

* A Blaschke–Potapov factor is a function of the form

\[\phi_\omega P + (1-P) \in \Inn(\C_+,B(\cH))\]

for some \(\omega \in \C_+\) and some finite rank projection \(P \in B(\cH)\).

* For \(\omega \in \C_+\) and \(v \in \cH\) we define the corresponding Blaschke–Potapov factor

\[\gls*{phiov} \coloneqq (\textbf{1} - P_{\C v}) + \phi_\omega P_{\C v}.\]

* A finite Blaschke–Potapov product is a function of the form

\[u \prod_{n=1}^N (\phi_{\omega_n} P_n + (1-P_n)) \in \Inn(\C_+,B(\cH))\]

with \(\omega_1,\dots,\omega_N \in \C_+\), finite rank projections \(P_1,\dots,P_N \in B(\cH)\) and \(u \in \U(\cH)\).

* The degree of a finite Blaschke–Potapov product

\[\phi = u \prod_{n=1}^N (\phi_{\omega_n} P_n + (1-P_n))\]

is defined as

\[\gls*{degBlaschke} \coloneqq \sum_{n=1}^N \dim P_n \cH \in \N_0.\]

* In the case \(\cH = \C\) Blaschke–Potapov factors and finite Blaschke–Potapov products are precisely Blaschke factors and finite Blaschke products (cf. def:Blaschke).

* For a finite dimensional subspace \(V \subeq \cH\), \(\omega \in \C_+\) and \(u \in \U(\cH)\) we have

\[u \left(\phi_\omega P_V + (1-P_V)\right) = \left(\phi_\omega P_{uV} + (1-P_{uV})\right) u.\]

This identity shows that the product of two finite Blaschke–Potapov products is again a finite  product.

* We will see that the degree is well defined in prop:FiniteCharacterization.

* By definition, for two finite Blaschke–Potapov products \(\phi\) and \(\psi\), one has

\[\deg(\phi \cdot \psi) = \deg(\phi) + \deg(\psi).\]

* If \(P\) is a finite rank projection and \((b_1,\dots,b_m)\) is an orthogonal basis of \(P\cH\), then

\[\phi_\omega P + (1-P) = \prod_{j=1}^m \phi_{\omega,b_j}. \]

Therefore every finite Blaschke–Potapov product \(\phi\) is of the form

\[\phi = u \prod_{n=1}^{\deg(\phi)} \phi_{\omega_n,v_n}\]

with \(\omega_1,\dots,\omega_{\deg(\phi)} \in \C_+\), \(v_1,\dots,v_{\deg(\phi)} \in \cH\) and \(u \in \U(\cH)\).

The following lemma will be useful later when analyzing finite Blaschke–Potapov products:

Let \(\cH\) be a complex Hilbert space, \(v \in \cH\) and \(\omega \in \C_+\). Then

\[\left(Q_\omega \cdot v\right)^{\perp_+} = M_{\phi_{\omega,v}} H^2(\C_+,\cH).\]

For \(v = 0\) both sides are equal to \(H^2(\C_+,\cH)\), so we can assume that \(v \neq 0\). One has \(f \in \left(Q_\omega \cdot v\right)^{\perp_+}\), if and only if

\[0 = \braket*{Q_\omega \cdot v}{f} = \braket*{v}{f(\omega)}.\]

Writing \(f = \tilde f \oplus f_v\) with \(\tilde f \in H^2(\C_+,\cH \ominus \C v)\) and \(f_v \in H^2(\C_+,\C v) \cong H^2(\C_+)\), this is equivalent to the function \(f_v\) being zero in \(\omega\), so, by prop:BlaschkeFactorization, \(f_v = \phi_\omega \cdot g\) for some \(g \in H^2(\C_+)\). Therefore

\[\left(Q_\omega \cdot v\right)^{\perp_+} = (\textbf{1} - P_{\C v})H^2(\C_+,\cH) + \phi_\omega P_{\C v}H^2(\C_+,\cH) = M_{\phi_{\omega,v}} H^2(\C_+,\cH). \qedhere\]

Using this lemma, we can connect \(M_{H^\infty}^*\)-stable subspaces to Blaschke–Potapov products:

Let \(\cH\) be a complex Hilbert space and \(V \subeq H^2(\C_+,\cH)\). Then \(V\) is a finite dimensional \(M_{H^\infty}^*\)-stable subspace, if and only if there exists a finite Blaschke–Potapov product \(\phi\) such that

\[V = \left(M_\phi H^2(\C_+,\cH)\right)^{\perp_+}\]

and in this case

\[\deg(\phi) = \dim V.\]

We first prove by induction over \(N \in \N_0\), that every \(M_{H^\infty}^*\)-stable subspace \(V \subeq H^2(\C_+,\cH)\) with \(\dim V = N\) is of the form \(V = \left(M_\phi H^2(\C_+,\cH)\right)^{\perp_+}\) for some finite Blaschke–Potapov product \(\phi\) with \(\deg(\phi) = N\). For \(N=0\) we have

\[V = \{0\} = \left(H^2(\C_+,\cH)\right)^{\perp_+} = \left(M_\textbf{1}H^2(\C_+,\cH)\right)^{\perp_+}\]

with \(\deg(\textbf{1}) = 0\).

We now assume that the statement is true for some \(N \in \N_0\) and let \(V \subeq H^2(\C_+,\cH)\) be a  subspace with \(\dim V = N +1\). Then, by lem:SpaceLokalDecomp, there exist \(\omega \in \C_+\) and \(v \in \cH \setminus \{0\}\) with \(Q_\omega \cdot v \in V\). Then, by lem:BlaschkeOrthogonal, we get

\[V^{\perp_+} \subeq (\C Q_\omega \cdot v)^{\perp_+} = M_{\phi_{\omega,v}} H^2(\C_+,\cH).\]

Using that \(M_{\phi_{\omega,v}}^{-1} = M_{\phi_{\omega,v}}^*\), this yields

\[M_{\phi_{\omega,v}}^* V^{\perp_+} \subeq H^2(\C_+,\cH).\]

Now, we set

\[W \coloneqq \left(M_{\phi_{\omega,v}}^* V^{\perp_+}\right)^{\perp_+}.\]

\begin{align*}
W &= P_+\left(M_{\phi_{\omega,v}}^* P_+V^\perp\right)^\perp = P_+M_{\phi_{\omega,v}}^* \left( P_+V^\perp\right)^\perp = P_+M_{\phi_{\omega,v}}^* \left(V + H^2(\C_-,\cH)\right) = P_+M_{\phi_{\omega,v}}^* V.
\end{align*}

So, for \(\phi \in H^\infty(\C_+)\), by lem:ProjectionEquation, one has

\begin{align*}
P_+ M_\phi^* W &= P_+M_\phi^* P_+ M_{\phi_{\omega,v}}^* V = P_+ M_{\phi_{\omega,v} \cdot \phi}^* V
\\&= P_+ M_{\phi \cdot \phi_{\omega,v}}^* V = P_+ M_{\phi_{\omega,v}}^* P_+M_\phi^* V \subeq P_+ M_{\phi_{\omega,v}}^* V = W
\end{align*}

and therefore \(W\) is \(M_{H^\infty}^*\)-stable. Further, using that

\[M_{\phi_{\omega,v}}^* Q_\omega(z) \cdot v = \frac{z-\overline{\omega}}{z-\omega} \cdot \frac 1{2\pi} \cdot \frac i{z-\overline{\omega}} \cdot v = \frac 1{2\pi} \cdot \frac i{z-\omega} \cdot v = Q_{\overline{\omega}}(z) \cdot v,\]

we have

\begin{align*}
M_{\phi_{\omega,v}}^* V = M_{\phi_{\omega,v}}^* (V \ominus \C Q_\omega \cdot v) \oplus M_{\phi_{\omega,v}}^* (\C Q_\omega \cdot v) = M_{\phi_{\omega,v}}^* (V \ominus \C Q_\omega \cdot v) \oplus \C Q_{\overline{\omega}} \cdot v.
\end{align*}

Therefore, since \(Q_{\overline{\omega}} \in H^2(\C_-)\) and

\[V \ominus \C Q_\omega \cdot v \subeq (\C Q_\omega \cdot v)^{\perp_+} = M_{\phi_{\omega,v}} H^2(\C_+,\cH)\]

we get

\begin{align*}
W = P_+ M_{\phi_{\omega,v}}^* V = P_+ \left(M_{\phi_{\omega,v}}^* (V \ominus \C Q_\omega \cdot v) \oplus \C Q_{\overline{\omega}} \cdot v\right) = M_{\phi_{\omega,v}}^* (V \ominus \C Q_\omega \cdot v),
\end{align*}

which implies

\[\dim W = \dim V - \dim \left(\C Q_\omega \cdot v\right) = (N+1) - 1 = N.\]

We therefore can apply the assumption of our induction and get a finite Blaschke–Potapov product \(\psi\) with \(\deg(\psi) = N\) such that
\(W = \left(M_\psi H^2(\C_+,\cH)\right)^{\perp_+}\). Setting \(\phi \coloneqq \phi_{\omega,v} \cdot \psi\), we get

\[\deg(\phi) = \deg(\phi_{\omega,v} \cdot \psi) = \deg(\psi) + \deg(\phi_{\omega,v}) = N + 1\]

\[V = \left(M_{\phi_{\omega,v}} W^{\perp_+}\right)^{\perp_+} = \left(M_{\phi_{\omega,v}} M_\psi H^2(\C_+,\cH)\right)^{\perp_+} = \left(M_\phi H^2(\C_+,\cH)\right)^{\perp_+}.\]

For the converse implication, we assume that \(\phi\) is a finite Blaschke–Potapov product, so

\[\phi = u \prod_{n=1}^N \phi_{\omega_n,v_n}\]

for some \(\omega_1,\dots,\omega_N \in \C_+\), \(v_1,\dots,v_N \in \cH\) and \(u \in \U(\cH)\). Setting

\[W_n \coloneqq u \prod_{j=1}^n \phi_{\omega_j,v_j} \cdot H^2(\C_+,\cH), \qquad n=0,\dots,N,\]

we have \(W_0 \supeq W_1 \supeq \dots \supeq W_N\) and

\begin{align*}
\dim(W_n^\perp \cap W_{n-1}) &= \dim \left(u \prod_{j=1}^{n-1} \phi_{\omega_j,v_j} \cdot \left(\left(\phi_{\omega_n,v_n} \cdot H^2(\C_+,\cH)\right)^\perp \cap H^2(\C_+,\cH)\right)\right)
\\&= \dim (M_{\phi_{\omega_n,v_n}} H^2(\C_+,\cH))^{\perp_+} = \dim \left(\C Q_{\omega_n} \cdot v_n\right) = 1,
\end{align*}

for \(n=1,\dots,N\), using lem:BlaschkeOrthogonal in the penultimate step. Then, for \(V \coloneqq \left(M_\phi H^2(\C_+,\cH)\right)^{\perp_+}\), we have

\[V = W_N^\perp \cap W_0 = \bigoplus_{n=1}^N \left(W_n^\perp \cap W_{n-1}\right)\]

and therefore

\begin{align*}
\dim V &= \sum_{n=1}^N \dim \left(W_n^\perp \cap W_{n-1}\right) = \sum_{n=1}^N 1 = N = \deg(\phi) < \infty.
\end{align*}

It remains to show that \(V\) is a \(M_{H^\infty}^*\)-stable subspace, but this follows by cor:stableCharakterizationRigid.

Finally, we will provide a way to calculate the degree of a finite Blaschke–Potapov product. For this we need the following defintion:

For a non-empty open subset \(U \subeq \C\), \(\xi \in U\) and \(f \in \cO(U \setminus \{\xi\})\) with Laurent series

\[f(z) = \sum_{n \in \Z} a_n (z-\xi)^n\]

we define the order of \(f\) in \(\xi\) by

\[\gls*{Ord}(f,\xi) \coloneqq \min\{n \in \Z: a_n \neq 0\}\]

if that minimum exists. Else we set

\[\mathrm{Ord}(f,\xi) \coloneqq -\infty\]

if \(a_n \neq 0\) for arbitrarily small \(n \in \Z\) and

\[\mathrm{Ord}(f,\xi) \coloneqq \infty\]

if \(a_n = 0\) for all \(n \in \Z\).

Further, for an open subset \(U \subeq \C_\infty\) with \(\infty \in U\) and \(f \in \cO(U \setminus \{\infty\})\), we set

\[\mathrm{Ord}(f,\infty) = \mathrm{Ord}\left(z \mapsto f\left(\frac 1z\right),0\right).\]

One can easily see that

\[\mathrm{Ord}(f \cdot g,\xi) = \mathrm{Ord}(f,\xi) + \mathrm{Ord}(g,\xi)\]

\[\mathrm{Ord}(f,\xi) \neq -\infty \neq \mathrm{Ord}(g,\xi).\]

This especially implies that

\[\mathrm{Ord}(c \cdot f,\xi) = \mathrm{Ord}(f,\xi)\]

for every constant \(c \in \C^\times\).

Let \(\cH\) be a finite dimensional complex Hilbert space and \(\phi\) be a finite Blaschke–Potapov product in \(H^\infty(\C_+,B(\cH))\). Then

\[\deg(\phi) = \sum_{\xi \in \C_+} \mathrm{Ord}(\det \circ \phi,\xi).\]

For a Blaschke–Potapov factor

\[\phi_\omega P + (1-P)\]

we have

\[\det(\phi_\omega(z) P + (1-P)) = \left(\frac{z-\omega}{z-\overline{\omega}}\right)^{\dim P\cH},\]

\[\mathrm{Ord}\left(\det \circ (\phi_\omega P + (1-P)),\xi\right) = \begin{cases} \dim P\cH & \text{if } \xi = \omega \\ 0 & \text{if } \xi \in \C_+ \setminus \{\omega\}\end{cases}\]

and therefore

\[\sum_{\xi \in \C_+} \mathrm{Ord}(\det \circ (\phi_\omega P + (1-P)),\xi) = \dim P\cH.\]

Now, for a finite Blaschke–Potapov product

\[\phi = u \prod_{n=1}^N (\phi_{\omega_n} P_n + (1-P_n))\]

and \(\xi \in \C_+\), we get

\begin{align*}
\mathrm{Ord}(\det \circ \phi,\xi) &= \mathrm{Ord}\left(\det \circ \left(u \prod_{n=1}^N (\phi_{\omega_n} P_n + (1-P_n))\right),\xi\right)
\\&= \mathrm{Ord}\left(\det(u) \cdot \prod_{n=1}^N (\det \circ (\phi_{\omega_n} P_n + (1-P_n))),\xi\right)
\\&= \sum_{n=1}^N \mathrm{Ord}\left(\det \circ (\phi_{\omega_n} P_n + (1-P_n)),\xi\right),
\end{align*}

using that \(\det(u) \in \T\). Therefore

\begin{align*}
\sum_{\xi \in \C_+}\mathrm{Ord}(\det \circ \phi,\xi) &= \sum_{\xi \in \C_+}\sum_{n=1}^N \mathrm{Ord}\left(\det \circ (\phi_{\omega_n} P_n + (1-P_n)),\xi\right)
\\&=\sum_{n=1}^N \sum_{\xi \in \C_+}\mathrm{Ord}\left(\det \circ (\phi_{\omega_n} P_n + (1-P_n)),\xi\right)
\\&=\sum_{n=1}^N \dim P_n \cH = \deg(\phi). \qedhere
\end{align*}

§.§.§ The kernel space

In this subsection, given a regular Pick function \(F \in \mathrm{Pick}(\C_+,B(\cH))_\R\), we will apply our results obtained in the last subsection to the kernel space \(\ker\left(\overline{\lambda}\textbf{1}-M_F^*\right)\) appearing in thm:kernelAdjointGeneral. We first prove that, in fact, \(\ker\left(\overline{\lambda}\textbf{1}-M_F^*\right)\) is a \(M_{H^\infty}^*\)-stable subspace:

Let \(\cH\) be a Hilbert space and \(F \in \mathrm{Pick}(\C_+,B(\cH))_\R\). Then, for every \(\lambda \in \C_+\) the space \(\ker\left(\overline{\lambda}\textbf{1}-M_F^*\right)\) is \(M_{H^\infty}^*\)-stable.

Let \(\phi \in H^\infty(\C_+)\) and \(g \in \ker\left(\overline{\lambda}\textbf{1}-M_F^*\right)\). Then, for \(f \in D(M_F)\), we have

\[\braket*{P_+M_\phi^*g}{(\lambda\textbf{1}-M_F)f} = \braket*{g}{M_\phi(\lambda\textbf{1}-M_F)f} = \braket*{g}{(\lambda\textbf{1}-M_F)M_\phi f} = \braket*{(\overline{\lambda}\textbf{1}-M_F^*)g}{M_\phi f} = 0,\]

where we use that \(M_\phi f \in D(F)\) with \(M_F(M_\phi f) = M_\phi M_Ff\). So \(P_+M_\phi^*g \in \ker\left(\overline{\lambda}\textbf{1}-M_F^*\right)\).

This lemma together with cor:stableCharakterizationRigid implies that there exists a function \(\phi \in H^\infty(\C_+,B(\cH))\) such that

\[\ker\left(\overline{\lambda}\textbf{1}-M_F^*\right) = \left(M_\phi H^2(\C_+,\cH)\right)^{\perp_+}.\]

We want to give an explicit formula for \(\phi\). For this we need the following lemmata:

Let \(\cH\) be a finite dimensional complex Hilbert space and \(F \in \mathrm{Pick}(\C_+,B(\cH))_\R\) be a regular Pick function. Then

\[\Phi_F: L^\infty(\R,\C) \to L^\infty(\R,B(\cH)), \quad f \mapsto f \circ F_*\]

is a well defined linear contraction.

For a function \(f \in \cL^\infty(\R,\C)\) the expression \(f \circ F_*\) makes sense as an element in \(L^\infty(\R,B(\cH))\), since for almost every \(x \in \R\) the boundary value \(F_*(x)\) is self-adjoint and therefore has real eigenvalues. Also it is clear that this map is linear and satisfies \(\left\lVert \Phi_F\right\rVert \leq 1\) since \(\left\lVert f \circ F_*\right\rVert_\infty \leq \left\lVert f\right\rVert_\infty\) for every \(f \in \cL^\infty(\R,\C)\). It remains to show that this map factorizes over the quotient map

\[q: \cL^\infty(\R,\C) \to \cL^\infty(\R,\C) \slash \cN = L^\infty(\R,B(\cH)),\]

where by \(\cN\) we denote the subspace of almost everywhere null functions. So let \(f \in \cN\). Then there is a Lebesgue zero-set \(M\) such that \(f\big|_{\R \setminus M} \equiv 0\) and

\[\{x \in \R: f \circ F_* \neq 0\} \subeq \{x \in \R: \Spec(F_*(x)) \cap M \neq \eset\} = D(F,M),\]

which, by thm:SpecNonDegZeroSet is a Lebesgue zero-set. Therefore \(f \circ F_* = 0 \in L^\infty(\R,B(\cH))\).

Let \(\cH,\cK\) be complex Hilbert spaces with \(\dim \cH < \infty\) and \(F \in \mathrm{Pick}(\C_+,B(\cH))_\R\) be regular. Further let

\[\psi: L^2(\R,\cH) \to L^2(\R,\cK)\]

be a unitary map such that

\[\psi \circ U^F_t = S_t \circ \psi \qquad \forall t \in \R,\]

where \(S_t \coloneqq M_{e^{it\mathrm{Id}}}\). Then, for every \(h \in L^\infty(\R,\C)\), one has

\[\psi \circ M_{h \circ F_*} = M_h \circ \psi.\]

The linear maps

\[L^\infty(\R,\C) \to B(L^2(\R,\cH)), \quad h \mapsto \psi^{-1} \circ M_h \circ \psi\]

\[L^\infty(\R,\C) \to B(L^2(\R,\cH)), \quad h \mapsto M_{h \circ F_*}\]

are both bounded, where for the boundedness of the second one we use lem:compPickBounded. Further they coincide on the subset \(\{e^{it \Id} : t \in \R\} \subeq L^\infty(\R,\C)\). Since these functions span a weakly dense subspace by prop:etDense, the two functions coincide, so, for every \(h \in L^\infty(\R,\C)\), we have

\[\psi^{-1} \circ M_h \circ \psi = M_{h \circ F_*}. \qedhere\]

Let \(\cH\) be a finite dimensional complex Hilbert space, \(F \in \mathrm{Pick}(\C_+,B(\cH))_\R\) be regular and \(\lambda \in \C_+\). Then

\[\ker\left(\overline{\lambda}\textbf{1}-M_F^*\right) = (M_{\phi_\lambda \circ F} H^2(\C_+,\cH))^{\perp_+}.\]

Let \(\cK\) be the multiplicity space of \((L^2(\R,\cH),H^2(\C_+,\cH),U^F)\) and \(\psi: L^2(\R,\cH) \to L^2(\R,\cK)\) be an isomorphism between \((L^2(\R,\cH),H^2(\C_+,\cH),U^F)\)  and \((L^2(\R,\cK),H^2(\C_+,\cK),S)\). Then, by lem:kernelAdjoint, we have

\[\psi \ker\left(\overline{\lambda}\textbf{1}-M_F^*\right) = Q_\lambda \cdot \cK,\]

so, by lem:BlaschkeOrthogonal and lem:firstCompRelation, we have

\begin{align*}
\ker\left(\overline{\lambda}\textbf{1}-M_F^*\right)^{\perp_+} &= \psi^{-1}(Q_\lambda \cdot \cK)^{\perp_+} = \psi^{-1} M_{\phi_\lambda} H^2(\C_+,\cK)
\\&= M_{\phi_\lambda \circ F} \psi^{-1} H^2(\C_+,\cK) = M_{\phi_\lambda \circ F} H^2(\C_+,\cH),
\end{align*}

which proves the equation since the kernel is a closed subspace.

Let \(\cH\) be a finite dimensional complex Hilbert space, \(F \in \mathrm{Pick}(\C_+,B(\cH))_\R\) be regular and \(\lambda \in \C_+\). Then, one has

\[\dim \ker\left(\overline{\lambda}\textbf{1}-M_F^*\right) < \infty,\]

if and only if \(\phi_\lambda \circ F\) is a finite Blaschke–Potapov product and in this case

\[\dim \ker\left(\overline{\lambda}\textbf{1}-M_F^*\right) = \deg(\phi_\lambda \circ F).\]

By lem:kernelShiftInvariant and prop:FiniteCharacterization we have that

\[\dim \ker\left(\overline{\lambda}\textbf{1}-M_F^*\right) < \infty,\]

if and only if there exists a finite Blaschke–Potapov product \(\phi\) with \(\deg(\phi) = \dim \ker\left(\overline{\lambda}\textbf{1}-M_F^*\right)\) such that

\[\ker\left(\overline{\lambda}\textbf{1}-M_F^*\right) = \left(M_\phi H^2(\C_+,\cH)\right)^{\perp_+}.\]

By prop:PickKernelForm we also have

\[\ker\left(\overline{\lambda}\textbf{1}-M_F^*\right) = \left(M_{\phi_\lambda \circ F} H^2(\C_+,\cH)\right)^{\perp_+},\]

so we get

\[M_\phi H^2(\C_+,\cH) = M_{\phi_\lambda \circ F} H^2(\C_+,\cH)\]

and therefore, by cor:H2EqualFunctions, we have

\[\phi_\lambda \circ F = \phi \cdot u\]

for some unitary operator \(u \in \U(\cH)\). This shows that

\[\dim \ker\left(\overline{\lambda}\textbf{1}-M_F^*\right) < \infty,\]

if and only if \(\phi_\lambda \circ F\) is a finite Blaschke–Potapov product with

\[\deg(\phi_\lambda \circ F) = \deg(\phi) = \dim \ker\left(\overline{\lambda}\textbf{1}-M_F^*\right). \qedhere\]

With these results we get a characterization of the multiplicity space of a ROPG infinitisimally generated by a Pick function:

Let \(\cH\) be a finite dimensional complex Hilbert space, \(F \in \mathrm{Pick}(\C_+,B(\cH))_\R\) be regular and let \(\cK\) be the multiplicity space of \((L^2(\R,\cH),H^2(\C_+,\cH),U^F)\). Then \(\dim \cK < \infty\), if and only if for one – or equivalently for all – \(\lambda \in \C_+\) the function \(\phi_\lambda \circ F\) is a finite Blaschke–Potapov product. In this case

\[\dim \cK = \deg(\phi_\lambda \circ F) = \sum_{\xi \in \C_+} \mathrm{Ord}(\det \circ \phi_\lambda \circ F,\xi) \quad \forall \lambda \in \C_+.\]

This follows immediately by cor:PickFiniteCharacterizationBlaschke and prop:BlaschkeDegFormula, since by thm:kernelAdjointGeneral we have

\[\cK \cong \ker\left(\overline{\lambda}\textbf{1}-M_F^*\right) \quad \forall \lambda \in \C_+. \qedhere\]

§.§.§ Holomorphic extensions

To avoid dealing with infinities, in the following we switch from the upper half plane \(\C_+\) to the unit disc \(\D\) via Cayley transform, so we consider the holomorphic map:

\[\gls*{h}: \C_+ \to \D, \quad z \mapsto \frac{z-i}{z+i}.\]

Setting \(f_\lambda \coloneqq \det \circ \phi_\lambda \circ F \circ h^{-1}: \D \to \C\) for a regular Pick function \(F \in \mathrm{Pick}(\C_+,B(\cH))_\R\) and \(\lambda \in \C_+\), we would like to write the sum appearing in prop:DimByOrd, using the argument principle, as

\[\sum_{\xi \in \C_+} \mathrm{Ord}(\det \circ \phi_\lambda \circ F,\xi) = \sum_{\omega \in \D} \mathrm{Ord}(f_\lambda,\omega) = \frac 1{2\pi i} \int_{f_\lambda \circ \gamma} \frac 1z \,dz\]

for some closed path \(\gamma\) in \(\C\) enclosing the closed unit disc \(\overline{\D}\). For this to be true, we would need that the function \(f_\lambda\) has a holomorphic extension to a simply connected domain containing the closed unit disc \(\overline{\D}\). We will later see that the right notion of extendability is the following:

Let \(\cH\) be a finite dimensional complex Hilbert space and \(\lambda \in \C_+\). We call \(F \in \mathrm{Pick}(\C_+,B(\cH))_\R\) , if there exists a finite set \(E_\lambda \subeq \C_-\) such that

\[\phi_\lambda \circ F: \C_+ \to B(\cH)\]

can be holomorphically extended to a function \(F_\lambda: \C_\infty \setminus E_\lambda \to B(\cH)\), such that, for every \(v,w \in \cH\), the matrix coeffizient

\[\C_\infty \setminus E_\lambda \ni z \mapsto \braket*{v}{F_\lambda(z)w} \in \C\]

is meromorphic. For a \(\lambda\)-extendable Pick function \(F\), we write \(\gls*{Elmin}\) for the minimal choice of \(E_\lambda\) and \(\gls*{Flmin}\) for the corresponding extension.

To classify the \(\lambda\)-extendable Pick functions, we need the following lemmata:
(cf. <cit.>)
Let \(\cH\) be a complex Hilbert space and let \(\phi: \C_+ \to B(\cH)\) be an inner function. Then \(\phi\) is a finite Blaschke–Potapov product, if and only if there is a finite Blaschke product \(\theta \in \Inn(\C_+)\) (cf. def:Blaschke) such that

\[M_\theta H^2(\C_+,\cH) \subeq M_\phi H^2(\C_+,\cH).\]

Let \(\cH\) be a finite dimensional Hilbert space and \(A \in B(\cH)\) be normal. Then, for every \(S \in B(\cH)\), one has

\[\Spec(A+S) \subeq \Spec(A) + U_{\leq \left\lVert S\right\rVert}(0).\]

\[\mu \in \C \setminus (\Spec(A) + U_{\leq \left\lVert S\right\rVert}(0)).\]

\[\dist(\mu,\Spec(A)) > \left\lVert S\right\rVert.\]

Since \(A\) and therefore \(A-\mu \textbf{1}\) is normal, we have

\begin{align*}
\left\lVert(A-\mu \textbf{1})^{-1}\right\rVert^{-1} &= \min\{|\lambda|: \lambda \in \Spec(A-\mu \textbf{1})\}
\\&= \min\{|\lambda-\mu|: \lambda \in \Spec(A)\} = \dist(\mu,\Spec(A)) > \left\lVert S\right\rVert,
\end{align*}

which implies that

\[(A -\mu \textbf{1}) + S = (A+S) - \mu \textbf{1}\]

is invertible and therefore \(\mu \in \C \setminus \Spec(A+S)\).

We now can formulate our theorem characterizing \(\lambda\)-extendable Pick functions using the following definition:

Let \(\cH\) be a finite dimensional complex Hilbert space. We call a Pick function \(F \in \mathrm{Pick}(\C_+,B(\cH))_\R\) rational, if
there are \(C \in S(\cH)\), \(D \in S(\cH)_+\) and \(A_1,\dots,A_m \in S(\cH)_+\)
such that

\[F(z) = C + zD + \sum_{j=1}^m \frac {1}{\lambda_j-z}A_j\]

and we call

\[\gls*{degPick} \coloneqq \rk(D) + \sum_{j=1}^m \rk(A_j)\]

the degree of \(F\). Further, we write \(\gls*{Rat}\) for the set of rational Pick functions and set

\[\deg(F) \coloneqq \infty\]

for all \(F \in \mathrm{Pick}(\C_+,B(\cH))_\R \setminus \mathrm{Rat}(\cH)\).

Let \(\cH\) be a finite dimensional complex Hilbert space, \(F \in \mathrm{Pick}(\C_+,B(\cH))_\R\) be regular and \(\lambda \in \C_+\). Then the following are equivalent:

* \(\phi_\lambda \circ F\) is a finite Blaschke–Potapov product.

* \(F\) is \(\lambda\)-extendable.

* \(F \in \mathrm{Rat}(\cH)\).

(a) \(\Rightarrow\) (b):

\[\phi_\lambda \circ F = u \prod_{n=1}^N (\phi_{\omega_n} P_n + (1-P_n))\]

with \(\omega_1,\dots,\omega_N \in \C_+\), projections \(P_1,\dots,P_N \in B(\cH)\) and \(u \in \U(\cH)\). All the functions \(\phi_{\omega_n}\) can be holomorphically extended to \(\C \setminus \{\overline{\omega_n}\}\), so setting \(E \coloneqq \{\overline{\omega_1},\dots,\overline{\omega_N}\} \subeq \C_-\), we get that \(\phi_\lambda \circ F\) can be holomorphically extended to a function on \(\C \setminus E\). To get the holomorphic extension in \(\infty\), we notice that

\[(\phi_\lambda \circ F)\left(\frac 1z\right) = u \prod_{n=1}^N \left(\frac{\frac 1z - \omega_n}{\frac 1z - \overline{\omega_n}} P_n + (1-P_n)\right) = u \prod_{n=1}^N \left(\frac{1 - \omega_n z}{1 - \overline{\omega_n}z} P_n + (1-P_n)\right),\]

which shows that \(\phi_\lambda \circ F\) can be holomorphically extended to \(\infty\) with the value

\[F_\lambda^{\mathrm{min}}(\infty) = u \prod_{n=1}^N \left(\frac{1 - \omega_n \cdot 0}{1 - \overline{\omega_n} \cdot 0} P_n + (1-P_n)\right) = u \prod_{n=1}^N \left(P_n + (1-P_n)\right) = u.\]

That, for every \(v,w \in \cH\), the matrix coefficient

\[\C_\infty \setminus E_\lambda \ni z \mapsto \braket*{v}{F(z)w} \in \C\]

is meromorphic, follows from the fact that all the functions \(\phi_{\omega_n}\), \(n=1,\dots,N\), are meromorphic.

(b) \(\Rightarrow\) (a):
Since \(F\) is \(\lambda\)-extendable, all the matrix coefficients of \(F_\lambda^{\mathrm{min}}\) are meromorphic and therefore rational functions. Since their poles are located in \(E_\lambda^{\mathrm{min}} \subeq \C_-\), we can write

\[F_\lambda^{\mathrm{min}}(z) = \left(\prod_{\omega \in E_\lambda^{\mathrm{min}}}(z-\omega)^{-m_\omega}\right) P(z),\]

where \(P\) is a matrix-valued function whose entries are polynomials and \(m_\omega \in \N\) for every \(\omega \in E_\lambda^{\mathrm{min}}\). Since \(F_\lambda^{\mathrm{min}}\) can be holomorphically extended to \(\infty\), we know that, for all \(v,w \in \cH\), the degree of the polynomial \(\C \ni z \mapsto \braket*{v}{P(z)w}\) is at most \(\sum_{\omega \in E_\lambda^{\mathrm{min}}}^N m_\omega\). This shows that for the function

\[g(z) = \left(\prod_{\omega \in E_\lambda^{\mathrm{min}}}(z-\overline{\omega})^{-m_\omega}\right) P(z)\]

one has \(g \in H^\infty(\C_-,B(\cH))\). Further, defining the Blaschke product

\[\theta \coloneqq \prod_{\omega \in E_\lambda^{\mathrm{min}}}\phi_{\overline{\omega}}^{m_\omega}\]

we have \(F_\lambda^{\mathrm{min}} = \theta g\). We now define \(h \in H^\infty(\C_+,B(\cH))\) by \(h(z) \coloneqq g(\overline{z})^*\). Then, for \(x \in \R\), we have

\[F_\lambda^{\mathrm{min}}(x) = \theta(x) h(x)^*.\]

Since \(F_\lambda^{\mathrm{min}}\) and \(\theta\) have unitary boundary values, also \(h\) has unitary boundary values and therefore \(\theta = F_\lambda^{\mathrm{min}} h\), which yields

\[M_\theta H^2(\C_+,\cH) = M_{F_\lambda^{\mathrm{min}}} M_h H^2(\C_+,\cH) \subeq M_{F_\lambda^{\mathrm{min}}} H^2(\C_+,\cH) = M_{\phi_\lambda \circ F} H^2(\C_+,\cH).\]

By lem:ratBlaschkePotapov this implies that \(\phi_\lambda \circ F\) is a finite Blaschke–Potapov product.

(b) \(\Rightarrow\) (c):
Since \(F\) is \(\lambda\)-extendable, all the matrix coefficients of \(F_\lambda^{\mathrm{min}}\) are meromorphic and therefore rational functions. This implies that also \(\det \circ \left(F_\lambda^{\mathrm{min}} - \textbf{1}\right)\) is a rational function and therefore the set

\[T \coloneqq \{x \in \R : 1 \in \Spec\left(F_\lambda^{\mathrm{min}}(x)\right)\}\]

is finite. For \(x \in \R \setminus T\), we have \(\phi_\lambda^{-1}\left(F_\lambda^{\mathrm{min}}(x)\right) \in B(\cH)\),
which implies that the limit

\[\lim_{\epsilon \downarrow 0} F(x+i \epsilon) = \phi_\lambda^{-1} \left(\lim_{\epsilon \downarrow 0} F_\lambda^{\mathrm{min}}(x+i \epsilon)\right) = \phi_\lambda^{-1}\left(F_\lambda^{\mathrm{min}}(x)\right)\]

exists. This implies that for every \(v \in \cH\) we have

\[\lim_{\epsilon \downarrow 0} \,\braket*{v}{F(x+i \epsilon)v} \neq \infty,\]

so the measure corresponding to the Pick function

\[\C_+ \ni z \mapsto \braket*{v}{F(z)v},\]

by thm:GePick is supported on the finite set \(T\). Together with <cit.> this implies that there are operators \((A_t)_{t \in T} \subeq S(\cH)_+\) and an operator \(C \in S(\cH)\) and an operator \(D \in S(\cH)_+\) such that

\[F(z) = C + zD + \sum_{t \in T}\frac 1{t-z}A_t,\]

which implies that \(F\) is a rational Pick function.

(c) \(\Rightarrow\) (b):
Let \(F \in \mathrm{Rat}(\cH)\), so

\[F(z) = C + zD + \sum_{j=1}^m \frac {1}{\lambda_j-z}A_j\]

for some \(C \in S(\cH)\), \(D \in S(\cH)_+\) and \(A_1,\dots,A_m \in S(\cH)_+\). By replacing \(F\) with \(F \circ m^{-1}\), where \(m\) is an automorphism of the Riemann sphere with \(m(\C_+) = \C_+\), chosen in a suitable way, for example as the Möbius transformation

\[m(z) = \frac 1{\xi-z} \quad \text{for } \xi \in \R \setminus \{\lambda_1,\dots,\lambda_n\},\]

we can without loss of generality assume that \(D=0\), so

\[F(z) = C + \sum_{j=1}^m \frac {1}{\lambda_j-z}A_j.\]

Then \(F\) can be holomorphically extended to a function

\[\tilde F: \C_\infty \setminus \{\lambda_1,\dots,\lambda_m\} \to B(\cH).\]

Now, for \(k \in \{1,\dots,m\}\), we define

\[F_k: \C_\infty \setminus \{\lambda_1,\dots,\lambda_{k-1},\lambda_{k+1},\lambda_m\} \to B(\cH), \quad z \mapsto C + \sum_{\substack{j \in \{1,\dots,m\} \\ j\neq k}} \frac {1}{\lambda_j-z}A_j\]

and set

\[K \coloneqq \max\left\{\left\lVert F_k(\lambda_k)\right\rVert : k=1,\dots,m\right\}.\]

Since the spectra \((\Spec(A_k))_{k=1,\dots,m}\) are all finite, there are \(a_1,\dots,a_m>0\) such that

\[\Spec(A_k) \subeq \{0\} \cup [a_k,\infty) \quad \forall k \in \{1,\dots,m\}.\]

We now choose \(\epsilon_1,\dots,\epsilon_m>0\) such that the sets

\[U_{\epsilon_1}(\lambda_1), \dots, U_{\epsilon_m}(\lambda_m)\]

are pairwise disjoint and

\[\left\lVert F_k(z) - F_k(\lambda_k)\right\rVert \leq 1 \quad \text{and} \quad \left|\frac 1{z-\lambda_k}\right|a_k \geq 2K+4\]

for every \(z \in U_{\epsilon_k}(\lambda_k) \setminus \{\lambda_k\}\). Then, for every \(z \in U_{\epsilon_k}(\lambda_k) \setminus \{\lambda_k\}\), one has

\[\left\lVert \tilde F(z) - \frac 1{z-\lambda_k}A_k\right\rVert = \left\lVert F_k(z)\right\rVert \leq \left\lVert F_k(\lambda_k)\right\rVert + \left\lVert F_k(z) - F_k(\lambda_k)\right\rVert \leq K + 1\]

\[\Spec\left(\frac 1{z-\lambda_k}A_k\right) \subeq \{0\} \cup \C \setminus U_{\leq \left|\frac 1{z-\lambda_k}\right|a_k}(0) \subeq \{0\} \cup \C \setminus U_{\leq 2K+4}(0).\]

Therefore, by lem:SpectrumPerturbation, using that \(A_k\) is self-adjoint and therefore \(\frac 1{z-\lambda_k}A_k\) is normal, we get

\begin{align*}
\Spec(\tilde F(z)) &\subeq \Spec\left(\frac 1{z-\lambda_k}A_k\right) + U_{\leq K+1}(0) \subeq \left(\{0\} \cup \C \setminus U_{\leq 2K+4}(0)\right) + U_{\leq K+1}(0)
\\&= U_{\leq K+1}(0) \cup \C \setminus U_{\leq (2K+4)-(K+1)}(0) = U_{\leq K+1}(0) \cup \C \setminus U_{\leq K+3}(0)
\\&\subeq \C \setminus U_1(-(K+2)i).
\end{align*}

This shows, that, denoting for \(\omega \in \C_+\) by \(\tilde \phi_\omega\) the holomorphic extension of the function \(\phi_\omega\) to \(\C_\infty \setminus \{\overline{\omega}\}\), the function \(\tilde \phi_{(K+2)i} \circ \tilde F\) is bounded on the open set

\[U \coloneqq \bigcup_{k=1}^m U_{\epsilon_k}(\lambda_k) \setminus \{\lambda_k\}.\]