Hilbert space $F^{2}(\boldsymbol{\mu})$ if $\varphi_{\lambda}$ extends to a
bounded linear functional on $F^{2}(\boldsymbol{\mu})$, i.e. there is $C>0$
such that

$|\varphi_{\lambda}(p)|=|p(\lambda)|\leq C\|p\|_{\boldsymbol{\mu}},\qquad
p\in{\mathbb{C}}\left<Z_{1},\ldots,Z_{n}\right>.$

Denote

${\mathcal{D}}(\boldsymbol{\mu}):=\\{\lambda\in{\mathbb{C}}^{n}:\
\varphi_{\lambda}\text{ is a bounded linear functional on
}F^{2}(\boldsymbol{\mu})\\}.$

######  Definition 4.6.

The joint point spectrum of $(L_{Z_{1}}^{*},\ldots,L_{Z_{n}}^{*})$ is the set
$\sigma_{p}(L_{Z_{1}}^{*},\ldots,L_{Z_{n}}^{*})$ of all tuples
$\zeta=(\zeta_{1},\ldots,\zeta_{n})\in{\mathbb{C}}^{n}$ with the property that
there is $z\in F^{2}(\boldsymbol{\mu})$, $z\neq 0$ such that
$L_{Z_{i}}^{*}z=\zeta_{i}z$ for any $i\in\\{1,\ldots,n\\}$. In this case, we
say that $z$ is a joint eigenvector for the operators
$L_{Z_{1}}^{*},\ldots,L_{Z_{n}}^{*}$.

Given a $k$-tuple
$\lambda=(\lambda_{1},\ldots,\lambda_{n})\in{\mathbb{C}}^{n}$ we set
$\lambda_{\alpha}:=\lambda_{j_{1}}\cdots\lambda_{j_{m}}$ if
$\alpha=g_{j_{1}}\cdots g_{j_{m}}\in{\mathbb{F}}_{n}^{+}$ and
$\lambda_{g_{0}}:=1$.

######  Theorem 4.7.

Let $\boldsymbol{\mu}=\\{\mu_{\beta}\\}_{\beta\in{\mathbb{F}}_{n}^{+}}$ be a
sequence of strictly positive real numbers with $\mu_{g_{0}}:=1$ and

$\sup_{\alpha\in{\mathbb{F}}_{n}^{+},i\in\\{1,\ldots,n\\}}\mu_{g_{i}\alpha}<\infty$

and let $F^{2}(\boldsymbol{\mu})$ be the Hilbert space of formal power series
in indeterminates $Z_{1},\ldots,Z_{n}$ associated with $\boldsymbol{\mu}$. If
$L_{Z}=(L_{Z_{1}},\ldots,L_{Z_{n}})$ is the left multi-shift associated with
$\boldsymbol{\mu}$, then the following statements hold.

  1. (i)

${\mathcal{D}}(\boldsymbol{\mu})=\left\\{(\lambda_{1},\ldots,\lambda_{n})\in{\mathbb{C}}^{n}:\
\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}\frac{|\lambda_{\alpha}|^{2}}{\boldsymbol{\mu}(\alpha,g_{0})^{2}}<\infty\right\\}=\sigma_{p}(L_{Z_{1}}^{*},\ldots,L_{Z_{n}}^{*}).$

  2. (ii)

If $\lambda\in{\mathcal{D}}(\boldsymbol{\mu})$ and
$f=\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}c_{\alpha}Z_{\alpha}$ is in the space
$F^{2}(\boldsymbol{\mu})$, then the series
$f(\lambda):=\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}c_{\alpha}\lambda_{\alpha}$
is absolutely convergent to $\varphi_{\lambda}(f)$ and

$|f(\lambda)|\leq\|f\|_{\boldsymbol{\mu}}\left(\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}\frac{|\lambda_{\alpha}|^{2}}{\boldsymbol{\mu}(\alpha,g_{0})^{2}}\right)^{1/2}.$

  3. (iii)

If $\lambda=(\lambda_{1},\ldots,\lambda_{n})\in{\mathbb{C}}^{n}$ and the
series
$\sum_{k=0}^{\infty}\sum_{\alpha\in{\mathbb{F}}_{n}^{+},|\alpha|=k}c_{\alpha}\lambda_{\alpha}$
is convergent for any element
$f=\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}c_{\alpha}Z_{\alpha}$ is
$F^{2}(\boldsymbol{\mu})$, then $\lambda\in{\mathcal{D}}(\boldsymbol{\mu})$.

  4. (iv)

If $\varphi=\sum_{\alpha\in{\mathbb{F}}_{n}}c_{\alpha}Z_{\alpha}$ is a bounded
free holomorphic function on the open ball of $B({\mathcal{H}})^{n}$ of radius
$\|L_{Z}\|$, then $\varphi\in{\mathcal{M}}^{\ell}(F^{2}(\boldsymbol{\mu}))$
and

$\|L_{\varphi}\|=\|\varphi(L_{Z_{1}},\ldots,L_{Z_{n}})\|\leq\sup\\{\|\varphi(X_{1},\ldots,X_{n})\|:\
\|(X_{1},\ldots,X_{n})\|<\|L_{Z}\|\\}.$

###### Proof.

If
$\lambda=(\lambda_{1},\ldots,\lambda_{n})\in{\mathcal{D}}(\boldsymbol{\mu})$,
then $\varphi_{\lambda}$ is a bounded linear functional on
$F^{2}(\boldsymbol{\mu})$ and the Riesz representation theorem implies the
existence of an element $k_{\lambda}\in F^{2}(\boldsymbol{\mu})$ such that
$\varphi_{\lambda}(f)=\left<f,k_{\lambda}\right>_{\boldsymbol{\mu}}$ for any
$f\in F^{2}(\boldsymbol{\mu})$. Since
$\left<Z_{\alpha},k_{\lambda}\right>_{\boldsymbol{\mu}}=\lambda_{\alpha}$ for
any $\alpha\in{\mathbb{F}}_{n}^{+}$, we deduce that
$k_{\lambda}=\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}\bar{\lambda}_{\alpha}\frac{1}{\boldsymbol{\mu}(\alpha,g_{0})^{2}}Z_{\alpha}$.
Since $k_{\lambda}\in F^{2}(\boldsymbol{\mu})$, we must have
$\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}\frac{|\lambda_{\alpha}|^{2}}{\boldsymbol{\mu}(\alpha,g_{0})^{2}}<\infty$.
The converse is obvious.

To prove the second equality in the part (i), let
$\lambda\in{\mathcal{D}}(\boldsymbol{\mu})$ and note that

$\begin{split}\left<f,L_{Z_{i}}^{*}k_{\lambda}\right>_{\boldsymbol{\mu}}&=\left<L_{Z_{i}}f,\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}\bar{\lambda}_{\alpha}b_{\alpha}Z_{\alpha}\right>_{\boldsymbol{\mu}}\\\
&=\lambda_{i}\left<Z_{i}f,k_{\lambda}\right>_{\boldsymbol{\mu}}=\left<Z_{i}f,\bar{\lambda}_{i},k_{\lambda}\right>_{\boldsymbol{\mu}}\end{split}$

for any $f\in F^{2}(\boldsymbol{\mu})$ and $i\in\\{1,\ldots,n\\}$.
Consequently, $L_{Z_{i}}^{*}k_{\lambda}=\bar{\lambda}_{i},k_{\lambda}$ which
shows that
$(\bar{\lambda}_{1},\ldots,\bar{\lambda}_{n})\in\sigma_{p}(L_{Z_{1}}^{*},\ldots,L_{Z_{n}}^{*})$
and
$(\lambda_{1},\ldots,\lambda_{n})\in\sigma_{p}(L_{Z_{1}}^{*},\ldots,L_{Z_{n}}^{*})$.

Conversely, assume that
$(\bar{\lambda}_{1},\ldots,\bar{\lambda}_{n})\in\sigma_{p}(L_{Z_{1}}^{*},\ldots,L_{Z_{n}}^{*})$
and let $h\in F^{2}(\boldsymbol{\mu})$, $h\neq 0$, with the property that
$L_{Z_{i}}^{*}h=\bar{\lambda}_{i}h$ for any $i\in\\{1,\ldots,n\\}$. Consider
the bounded linear functional $\psi:F^{2}(\boldsymbol{\mu})\to{\mathbb{C}}$
given by $\psi(f):=\left<f,wh\right>_{\boldsymbol{\mu}}$ with
$w\in{\mathbb{C}}$, $w\neq 0$, to be determined. Note that

$\begin{split}\psi(Z_{i}f)=\left<Z_{i}f,wh\right>_{\boldsymbol{\mu}}=\left<f,wL_{Z_{i}}^{*}h\right>_{\boldsymbol{\mu}}=\lambda_{i}\left<f,wh\right>_{\boldsymbol{\mu}}=\lambda_{i}\psi(f)\end{split}$

for any $i\in\\{1,\ldots,n\\}$ which implies
$\psi(Z_{\alpha})=\lambda_{\alpha}\psi(1)$ for any
$\alpha\in{\mathbb{F}}_{n}^{+}$ with $|\alpha|\geq 1$. Since $\psi\neq 0$, we
must have $\psi(1)\neq 0$. Setting
$w:=\frac{1}{\overline{\left<1,h\right>_{\boldsymbol{\mu}}}}$, we have
$\psi(1)=1$ and, consequently, $\psi(p)=p(\lambda)$ for any
$p\in{\mathbb{C}}\left<Z_{1},\ldots,Z_{n}\right>$. Therefore,
$(\lambda_{1},\ldots,\lambda_{n})\in{\mathcal{D}}(\boldsymbol{\mu})$ which
completes the proof of part (i).

To prove part (ii), note that
$g:=\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}|c_{\alpha}|Z_{\alpha}$ is also in
$F^{2}(\boldsymbol{\mu})$. If $\lambda\in{\mathcal{D}}(\boldsymbol{\mu})$,
then, setting $g_{m}:=\sum_{\alpha\in{\mathbb{F}}_{n}^{+},|\alpha|\leq
m}|c_{\alpha}|Z_{\alpha}$, we deduce that
$g_{m}(\lambda)=\varphi_{\lambda}(g_{m})=\left<g_{m},k_{\lambda}\right>_{\boldsymbol{\mu}}\to\varphi_{\lambda}(g)$
as $m\to\infty$. Consequently, the series
$\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}|c_{\alpha}|\lambda_{\alpha}$ is
convergent for any $\lambda\in{\mathcal{D}}(\boldsymbol{\mu})$. Since
$(\lambda_{1}e^{i\theta_{1}},\ldots,\lambda_{n}e^{i\theta_{n}})\in{\mathcal{D}}(\boldsymbol{\mu})$
for any $\theta_{i}\in{\mathbb{R}}$, we also deduce that
$\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}|c_{\alpha}||\lambda_{\alpha}|$ is
convergent for any $\lambda\in{\mathcal{D}}(\boldsymbol{\mu})$. As above, we
deduce that, for any $\lambda\in{\mathcal{D}}(\boldsymbol{\mu})$,

$\sum_{\alpha\in{\mathbb{F}}_{n}^{+},|\alpha|\leq
m}c_{\alpha}\lambda_{\alpha}=\varphi_{\lambda}\left(\sum_{\alpha\in{\mathbb{F}}_{n}^{+},|\alpha|\leq
m}c_{\alpha}\lambda_{\alpha}\right)\to\varphi_{\lambda}(f)=\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}c_{\alpha}\lambda_{\alpha}$

as $m\to\infty$. Since
$f(\lambda)=\varphi_{\lambda}(f)=\left<f,k_{\lambda}\right>_{\boldsymbol{\mu}}$
for any $f\in F^{2}(\boldsymbol{\mu})$ and
$\|k_{\lambda}\|_{\boldsymbol{\mu}}=\left(\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}\frac{|\lambda_{\alpha}|^{2}}{\boldsymbol{\mu}(\alpha,g_{0})^{2}}\right)^{1/2}$,
the Cauchy-Schwartz inequality implies

$|f(\lambda)|\leq\|f\|_{\boldsymbol{\mu}}\left(\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}\frac{|\lambda_{\alpha}|^{2}}{\boldsymbol{\mu}(\alpha,g_{0})^{2}}\right)^{1/2},\qquad\lambda\in{\mathcal{D}}(\boldsymbol{\mu}),$

for any $f\in F^{2}(\boldsymbol{\mu})$.

Now, we prove part (iii). Assume that
$\lambda=(\lambda_{1},\ldots,\lambda_{n})\in{\mathbb{C}}^{n}$ and the series
$\sum_{k=0}^{\infty}\sum_{\alpha\in{\mathbb{F}}_{n}^{+},|\alpha|=k}c_{\alpha}\lambda_{\alpha}$
is convergent for any
$f=\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}c_{\alpha}Z_{\alpha}$ is
$F^{2}(\boldsymbol{\mu})$. Define the linear functional on
$F^{2}(\boldsymbol{\mu})$ by setting
$\varphi_{\lambda}(f):=\sum_{k=0}^{\infty}\sum_{\alpha\in{\mathbb{F}}_{n}^{+},|\alpha|=k}c_{\alpha}\lambda_{\alpha}$.
For each $m\in{\mathbb{N}}$, set
$k^{(m)}_{\lambda}:=\sum_{\alpha\in{\mathbb{F}}_{n}^{+},|\alpha|\leq
m}\bar{\lambda}_{\alpha}\frac{1}{\boldsymbol{\mu}(\alpha,g_{0})^{2}}Z_{\alpha}$.
Note that

$\left<f,k^{(m)}_{\lambda}\right>_{\boldsymbol{\mu}}=\sum_{\alpha\in{\mathbb{F}}_{n}^{+},|\alpha|\leq
m}c_{\alpha}\lambda_{\alpha}\to\varphi_{\lambda}(f)$

as $m\to\infty$. Applying the uniform boundedness principle we deduce that
$k_{\lambda}:=\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}\bar{\lambda}_{\alpha}\frac{1}{\boldsymbol{\mu}(\alpha,g_{0})^{2}}Z_{\alpha}$
is in the Hilbert $F^{2}(\boldsymbol{\mu})$. Due to part (i), we conclude that
$\lambda\in{\mathcal{D}}(\boldsymbol{\mu})$.

To prove part (iv), note that
$\psi:=\sum_{\alpha\in{\mathbb{F}}_{n}}c_{\alpha}\|L_{Z}\|^{|\alpha|}Z_{\alpha}$
is a bounded free holomorphic function on the open unit ball of
$B({\mathcal{H}})^{n}$ and, consequently,
$\Psi(S_{1},\ldots,S_{n})=\sum_{\alpha\in{\mathbb{F}}_{n}}c_{\alpha}\|L_{Z}\|^{|\alpha|}S_{\alpha}$
is the Fourier representation of a unique element $\Psi(S_{1},\ldots,S_{n})$
in the Hardy algebra $F^{\infty}(H_{n})$. Consider the weighted shift
$\Gamma:=(\frac{1}{\|L_{Z}\|}L_{Z_{1}},\ldots,\frac{1}{\|L_{Z}\|}L_{Z_{n}})$
and note that $\|\Gamma\|=1$. According to Theorem 1.12, $\Gamma$ is a pure
row contraction. Due to the $F^{\infty}(H_{n})$-functional calculus for pure
row contractions [42], we deduce that
$\|\psi(\Gamma_{1},\ldots\Gamma_{n})\|\leq\|\psi(S_{1},\ldots,S_{n})\|$.
Consequently, we have

$\begin{split}\|L_{\varphi}\|&=\|\varphi(L_{Z_{1}},\ldots,L_{Z_{n}})=\|\psi(\Gamma_{1},\ldots\Gamma_{n})\|\\\
&\leq\|\psi(S_{1},\ldots,S_{n})\|=\sup_{\|X\|<\|L_{Z}\|}\|\varphi(X_{1},\ldots,X_{n})\|.\end{split}$

The proof is complete. ∎

Let $\boldsymbol{\mu}=\\{\mu_{\beta}\\}_{\beta\in{\mathbb{F}}_{n}^{+}}$ be a
sequence of strictly positive real numbers satisfying relations (4.1) and
(4.2) and let $W=(W_{1},\ldots,W_{n})$ and
$\Lambda=(\Lambda_{1},\ldots,\Lambda_{n})$ be the associated weighted multi-
shifs. According to Theorem 4.7, we have a complete description of the joint
invariant subspaces of $W_{1},\ldots,W_{n}$ of co-dimension one. However, we
don’t have such a description for all the invariant subspaces. We remark that
a Beurling type description of all the invariant subspaces under
$W_{1},\ldots,W_{n}$ was obtained for certain particular classes of weights
(see [47], [48], [49], [50]).

######  Proposition 4.8.

There are no proper joint invariant subspaces ${\mathcal{M}}$ and
${\mathcal{N}}$ under $W_{1},\ldots,W_{n}$ such that
$F^{2}(H_{n})={\mathcal{M}}\oplus{\mathcal{N}}$ (direct sum).

###### Proof.

Assume that the decomposition above holds. Then there a nontrivial idempotent
$P\in B(F^{2}(H_{n}))$, i.e. $P^{2}=P$, $P\neq I$, $P\neq 0$, such that
$PW_{i}=W_{i}P$ for any $i\in\\{1,\ldots,n\\}$. Consequently,
$P=\psi(\Lambda_{1},\ldots,\Lambda_{n})\in R^{\infty}(\boldsymbol{\mu})$.
Since $\psi(\Lambda_{1},\ldots,\Lambda_{n})\neq 0$, one can easily show that
it is injective. The proof is similar to that corresponding to the particular
case when $\mu_{\beta}=1$ for any $\beta\in{\mathbb{F}}_{n}^{+}$ (see [15]).
Consequently, the relation
$\psi(\Lambda_{1},\ldots,\Lambda_{n})(\psi(\Lambda_{1},\ldots,\Lambda_{n})-I)=0$
implies $\psi(\Lambda_{1},\ldots,\Lambda_{n})=I$, which is a contradiction.
The proof is complete. ∎

######  Proposition 4.9.

Let $W=(W_{1},\ldots,W_{n})\in B(F^{2}(H_{n}))$ be the weighted left multi-
shift associated with
$\boldsymbol{\mu}=\\{\mu_{\beta}\\}_{\beta\in{\mathbb{F}}_{n}^{+},|\beta|\geq
1}$. Then the following statements hold.

  1. (i)

The joint point spectrum $\sigma_{p}(W_{1}^{*},\ldots,W_{n}^{*})$ coincides
with the set

$\\{\lambda:=(\lambda_{1},\ldots,\lambda_{n})\in{\mathbb{C}}^{n}:\
\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}\frac{|\lambda_{\alpha}|^{2}}{{\boldsymbol{\mu}(\alpha,g_{0})^{2}}}<\infty\\}.$

and the joint eigenvectors for the operators $W_{1}^{*},\ldots,W_{n}^{*}$ are
precisely the vectors

$z_{\lambda}:=\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}\frac{\bar{\lambda}_{\alpha}}{{\boldsymbol{\mu}(\alpha,g_{0})}}e_{\alpha},$

where
$\lambda:=(\lambda_{1},\ldots,\lambda_{n})\in\sigma_{p}(W_{1}^{*},\ldots,W_{n}^{*})$.
In this case, we have

$W_{i}^{*}z_{\lambda}=\bar{\lambda}_{i}z_{\lambda},\qquad
i\in\\{1,\ldots,n\\}.$

  2. (ii)

The joint point spectrum $\sigma_{p}(W_{1}^{*},\ldots,W_{n}^{*})$ is a
complete Reinhardt set in ${\mathbb{C}}^{n}$ containing the origin and

$\sigma_{p}(W_{1}^{*},\ldots,W_{n}^{*})\subset\\{(\lambda_{1},\ldots,\lambda_{n})\in{\mathbb{C}}^{n}:\
(|\lambda_{1}|^{2}+\cdots+|\lambda_{n}|^{2})^{1/2}\leq r(W)\\},$

where $r(W)$ is the joint spectral radius of $W=(W_{1},\ldots,W_{n})$.

  3. (iii)

If $\lambda=(\lambda_{1},\ldots,\lambda_{n})\in{\mathbb{C}}^{n}$ is such that
$\|\lambda\|=\|W\|$, then
$\lambda\notin\sigma_{p}(W_{1}^{*},\ldots,W_{n}^{*})$.

  4. (iv)

The joint point spectrum $\sigma_{p}(W_{1}^{*},\ldots,W_{n}^{*})$ contains a
ball centered at zero if and only if

$\limsup_{k\to\infty}\left(\sum_{\alpha\in{\mathbb{F}}_{n}^{+},|\alpha|=k}\frac{1}{{\boldsymbol{\mu}(\alpha,g_{0})^{2}}}\right)^{1/k}<\infty.$

In particular, the later relation holds if the weights are bounded above from
zero.

###### Proof.

As mentioned before, the operator $U:F^{2}(H_{n})\to F^{2}(\boldsymbol{\mu})$,
defined by
$U(e_{\alpha}):=\frac{1}{\boldsymbol{\mu}(\alpha,g_{0})}Z_{\alpha}$,
$\alpha\in{\mathbb{F}}_{n}^{+}$, is unitary and $UW_{i}=L_{Z_{i}}U,\
i\in\\{1,\ldots,n\\},$ where $W=(W_{1},\ldots,W_{n})$ is the weighted left
multi-shift associated with the weights
$\\{\mu_{\beta}\\}_{\beta\in{\mathbb{F}}_{n}^{+}}$. Now item (i) can be
extracted from Theorem 4.7. For the benefit of the reader, we present another
proof.

Let $z=\sum_{\beta\in{\mathbb{F}}_{n}^{+}}c_{\beta}e_{\beta}\in F^{2}(H_{n})$,
$z\neq 0$, and assume that $W_{i}^{*}z=\bar{\lambda}_{i}z$ for any
$i\in\\{1,\ldots,n\\}$ for some
$(\lambda_{1},\ldots,\lambda_{n})\in{\mathbb{C}}^{n}$. Using the definition of
the weighted left multi-shift, we deduce that

$\begin{split}c_{\alpha}&=\left<z,e_{\alpha}\right>=\left<z,e_{\alpha}\right>=\left<z,\frac{1}{{\boldsymbol{\mu}(\alpha,g_{0})}}W_{\alpha}1\right>\\\
&=\frac{1}{{\boldsymbol{\mu}(\alpha,g_{0})}}\left<W_{\alpha}^{*}z,1\right>=\frac{1}{{\boldsymbol{\mu}(\alpha,g_{0})}}\bar{\lambda}_{\alpha}\left<z,1\right>=\frac{c_{g_{0}}\bar{\lambda}_{\alpha}}{{\boldsymbol{\mu}(\alpha,g_{0})}}\end{split}$

for any $\alpha\in{\mathbb{F}}_{n}^{+}$. Consequently,
$z=c_{g_{0}}\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}\frac{\bar{\lambda}_{\alpha}}{{\boldsymbol{\mu}(\alpha,g_{0})}}e_{\alpha}\in
F^{2}(H_{n})$, which implies

$\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}\frac{|\lambda_{\alpha}|^{2}}{{\boldsymbol{\mu}(\alpha,g_{0})^{2}}}<\infty.$

Conversely, assume that
$\lambda=(\lambda_{1},\ldots,\lambda_{n})\in{\mathbb{C}}^{n}$ satisfies the
later relation. Then
$z_{\lambda}:=\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}\frac{\bar{\lambda}_{\alpha}}{{\boldsymbol{\mu}(\alpha,g_{0})}}e_{\alpha}$
is in the full Fock space $F^{2}(H_{n})$. Using the fact that
$\boldsymbol{\mu}(g_{i}\gamma,g_{0})=\mu_{g_{i}\gamma}\boldsymbol{\mu}(\gamma,g_{0})$,
we obtain

$\begin{split}W_{i}^{*}z_{\lambda}&=\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}\frac{\bar{\lambda}_{\alpha}}{{\boldsymbol{\mu}(\alpha,g_{0})}}W_{i}^{*}e_{\alpha}=\sum_{\gamma\in{\mathbb{F}}_{n}^{+}}\frac{\bar{\lambda}_{g_{i}\gamma}}{{\boldsymbol{\mu}(g_{i}\gamma,g_{0})}}\mu_{g_{i}\gamma}e_{\gamma}\\\
&=\bar{\lambda}_{i}\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}\frac{\bar{\lambda}_{\alpha}}{{\boldsymbol{\mu}(\alpha,g_{0})}}e_{\alpha}=\bar{\lambda}_{i}z_{\lambda}\end{split}$

for any $i\in\\{1,\ldots,n\\}$. This completes the proof of item (i).

To prove part (ii), fix
$\lambda:=(\lambda_{1},\ldots,\lambda_{n})\in\sigma_{p}(W_{1}^{*},\ldots,W_{n}^{*})$
and note that

$\begin{split}\left<\sum_{|\alpha|=k}W_{\alpha}W_{\alpha}^{*}z_{\lambda},z_{\lambda}\right>&=\sum_{|\alpha|=k}\|W_{\alpha}^{*}z_{\lambda}\|^{2}\\\
&=\|z_{\lambda}\|^{2}\sum_{|\alpha|=k}|\lambda_{\alpha}|^{2}=\|z_{\lambda}\|^{2}(|\lambda_{1}|^{2}+\cdots+|\lambda_{n}|^{2})^{k}\end{split}$

and, consequently,

$(|\lambda_{1}|^{2}+\cdots+|\lambda_{n}|^{2})^{1/2}\leq\left\|\sum_{|\alpha|=k}W_{\alpha}W_{\alpha}^{*}\right\|^{1/2k}$

for any $k\in{\mathbb{N}}$. Hence, we deduce that
$(|\lambda_{1}|^{2}+\cdots+|\lambda_{n}|^{2})^{1/2}\leq r(W)$. The fact that
$\sigma_{p}(W_{1}^{*},\ldots,W_{n}^{*})$ is a complete Reinhardt set in
${\mathbb{C}}^{n}$ containing the origin is obvious due to part (i).

To prove item (iii), let
$\lambda:=(\lambda_{1},\ldots,\lambda_{n})\in{\mathbb{C}}^{n}$. Since

$\left\|\sum_{\beta\in{\mathbb{F}}_{n}^{+},|\beta|=k}W_{\beta}W_{\beta}^{*}\right\|=\sup_{\alpha,\beta\in{\mathbb{F}}_{n}^{+},|\beta|=k}\boldsymbol{\mu}(\beta,\alpha)^{2}\geq\boldsymbol{\mu}(\beta,g_{0})^{2}$

we deduce that

$\begin{split}\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}\frac{|\lambda_{\alpha}|^{2}}{{\boldsymbol{\mu}(\alpha,g_{0})^{2}}}&\geq\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}\frac{|\lambda_{\alpha}|^{2}}{\left\|\sum_{\alpha\in{\mathbb{F}}_{n}^{+},|\alpha|=k}W_{\alpha}W_{\alpha}^{*}\right\|}\\\
&\geq\sum_{k=0}^{\infty}\frac{(|\lambda_{1}|^{2}+\cdots+|\lambda_{n}|^{2})^{k}}{\left\|\sum_{i=1}^{n}W_{i}W_{i}^{*}\right\|^{k}}\end{split}$

If $\|\lambda\|=\|W\|$, the later series is divergent, which proves that
$\lambda\notin\sigma_{p}(W_{1}^{*},\ldots,W_{n}^{*})$.

If the joint point spectrum $\sigma_{p}(W_{1}^{*},\ldots,W_{n}^{*})$ contains
a ball centered at zero, then there is $r_{0}\in(0,1)$ such that

$\sum_{k=0}^{\infty}\left(\sum_{|\alpha|=k}\frac{1}{{\boldsymbol{\mu}(\alpha,g_{0})^{2}}}\right)z^{k}\leq\sum_{k=0}^{\infty}\sum_{|\alpha|=k}\frac{r_{0}^{|\alpha|}}{{\boldsymbol{\mu}(\alpha,g_{0})^{2}}}<\infty$

for any complex number $z$ with $|z|\leq r_{0}$. Standard arguments concerning
the radius of convergence of a power series show that

$\limsup_{k\to\infty}\left(\sum_{\alpha\in{\mathbb{F}}_{n}^{+},|\alpha|=k}\frac{1}{{\boldsymbol{\mu}(\alpha,g_{0})^{2}}}\right)^{1/k}\leq\frac{1}{r_{0}}.$

Conversely, assume that the later relation holds for some $r_{0}>0$. Then
there exists $t_{0}>0$ such that
$\sum_{k=0}^{\infty}\left(\sum_{|\alpha|=k}\frac{1}{{\boldsymbol{\mu}(\alpha,g_{0})^{2}}}\right)t_{0}^{k}<\infty$.
Now, it is clear that, for any
$\lambda=(\lambda_{1},\ldots,\lambda_{n})\in{\mathbb{C}}^{n}$ with
$|\lambda_{i}|\leq t_{0}$, we have
$\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}\frac{|\lambda_{\alpha}|^{2}}{{\boldsymbol{\mu}(\alpha,g_{0})^{2}}}<\infty$,
which completes the proof of item (iv).

Now, assume that there exists $M>0$ such that $\mu_{\alpha}\geq M$ for any
$\alpha\in{\mathbb{F}}_{n}^{+}$. Then and $\boldsymbol{\mu}(\alpha,g_{0})\geq
M^{k}$ if $|\alpha|=k$ and

$\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}\frac{|\lambda_{\alpha}|^{2}}{{\boldsymbol{\mu}(\alpha,g_{0})^{2}}}\leq\sum_{k=0}^{\infty}\frac{1}{M^{2k}}(|\lambda_{1}|^{2}+\cdots+|\lambda_{n}|^{2})^{k}<\infty$

for any $(\lambda_{1},\ldots,\lambda_{k})$ with
$|\lambda_{1}|^{2}+\cdots+|\lambda_{n}|^{2}<M^{2}$. Using item (i), we
complete the proof. ∎

We remark that, in the particular case when $\mu_{\beta}=1$, part (i) of
Proposition 4.9 was proved in [4] and [15].

######  Example 4.10.

Let $\mu_{\beta}=1$ for any $\beta\in{\mathbb{F}}_{n}^{+}$. Then
$\boldsymbol{\mu}(\alpha,g_{0})=1$ and condition
$\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}\frac{|\lambda_{\alpha}|^{2}}{{\boldsymbol{\mu}(\alpha,g_{0})^{2}}}<\infty$
is equivalent to
$\sum_{k=0}^{\infty}(|\lambda_{1}|^{2}+\cdots+|\lambda_{n}|^{2})^{k}<\infty$,
which is equivalent to $|\lambda_{1}|^{2}+\cdots+|\lambda_{n}|^{2}<1$. In this
case we have

$\sigma_{p}(W_{1}^{*},\ldots,W_{n}^{*})={\mathbb{B}}_{n}=\\{\lambda\in{\mathbb{C}}^{n}:\
\|\lambda\|_{2}<1\\}.$

######  Example 4.11.

Let $\mu_{\beta}:=\left(\frac{|\beta|+1}{|\beta|}\right)^{2}$ for any
$\beta\in{\mathbb{F}}_{n}^{+}$ with $|\beta|\geq 1$, and $\mu_{g_{0}}:=1$. In
this case, we have $\boldsymbol{\mu}(\alpha,g_{0})=(|\alpha|+1)^{2}$ for any
$\alpha\in{\mathbb{F}}_{n}^{+}$, which implies

$\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}\frac{|\lambda_{\alpha}|^{2}}{{\boldsymbol{\mu}(\alpha,g_{0})^{2}}}=\sum_{k=0}^{\infty}\frac{(|\lambda_{1}|^{2}+\cdots+|\lambda_{n}|^{2})^{k}}{(k+1)^{4}}.$

It is easy to see that

$\sigma_{p}(W_{1}^{*},\ldots,W_{n}^{*})=\overline{{\mathbb{B}}}_{n}.$

######  Example 4.12.

Let $\mu_{\beta}:=\left(\frac{1}{|\beta|+1}\right)^{2}$ for any
$\beta\in{\mathbb{F}}_{n}^{+}$. Then
$\boldsymbol{\mu}(\alpha,g_{0})=\frac{1}{(|\alpha|+1)!}$ and

$\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}\frac{|\lambda_{\alpha}|^{2}}{{\boldsymbol{\mu}(\alpha,g_{0})^{2}}}=\sum_{k=0}^{\infty}((k+1)!)^{2}(|\lambda_{1}|^{2}+\cdots+|\lambda_{n}|^{2})^{k}<\infty$

if and only if $\lambda_{1}=\cdots\lambda_{n}=0$. Therefore,

$\sigma_{p}(W_{1}^{*},\ldots,W_{n}^{*})=\\{0\\}.$

We recall that the joint right Harte spectrum $\sigma_{r}(T_{1},\ldots,T_{n})$
of an $n$-tuple $(T_{1},\ldots,T_{n})\in B({\mathcal{H}})^{n}$ is the set of
all $n$-tuples $(\lambda_{1},\ldots,\lambda_{n})$ of complex numbers such that
the right ideal of $B({\mathcal{H}})$ generated by the operators
$\lambda_{1}I-T_{1},\ldots,\lambda_{n}I-T_{n}$ does not contain the identity
operator. The joint left Harte spectrum
$\sigma_{l}(T^{*}_{1},\ldots,T^{*}_{n})$ is the complex conjugate of
$\sigma_{r}(T_{1},\ldots,T_{n})$. We recall [43] that
$(\lambda_{1},\ldots,\lambda_{n})\notin\sigma_{r}(T_{1},\ldots,T_{n})$ if and
only if there exists $\delta>0$ such that
$\sum\limits_{i=1}^{n}(\lambda_{i}I-T_{i})(\overline{\lambda}_{i}I-T_{i}^{*})\geq\delta
I$.

The following result was proved in [24] in a more general case. We include a
proof in our particular setting.

######  Corollary 4.13.

$\sigma_{p}(W_{1}^{*},\ldots,W_{n}^{*})\subset\sigma_{l}(W^{*}_{1},\ldots,W^{*}_{n})$.

###### Proof.

Let
$\mu=(\mu_{1},\ldots,\mu_{n})\in{\mathcal{D}}(\boldsymbol{\mu})=\sigma_{p}(W_{1}^{*},\ldots,W_{n}^{*})$
and assume that there is $\delta>0$ such that

$\sum_{i=1}^{n}\|(W_{i}-\mu_{i}I)^{*}h\|^{2}\geq\delta\|h\|^{2}\quad\text{ for
all }\ h\in F^{2}(H_{n}).$

Taking
$h=z_{\lambda}:=\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}\frac{\bar{\lambda}_{\alpha}}{{\boldsymbol{\mu}(\alpha,g_{0})}}e_{\alpha}$,
$\lambda\in{\mathcal{D}}(\boldsymbol{\mu})$, in the inequality above, and
using Theorem 4.9, we obtain

$\sum_{i=1}^{n}|\lambda_{i}-\mu_{i}|^{2}\geq\delta\quad\text{ for all }\
\lambda\in{\mathcal{D}}(\boldsymbol{\mu}),$

which is a contradiction. Due to the remarks preceding the corollary, we
deduce that
$\mu=(\mu_{1},\ldots,\mu_{n})\in\sigma_{l}(W^{*}_{1},\ldots,W^{*}_{n})$, which
completes the proof. ∎

######  Theorem 4.14.

A map $\Phi:F^{\infty}(\boldsymbol{\mu})\to{\mathbb{C}}$ is a
$w^{*}$-continuous multiplicative linear functional if and only if there
exists $\lambda\in{\mathcal{D}}(\boldsymbol{\mu})$ such that

$\Phi(\psi(W))=\psi(\lambda),\qquad\psi(W)\in F^{\infty}(\boldsymbol{\mu}).$

In this case, we have

$\psi(\lambda)=\left<\psi(W)1,z_{\lambda}\right>=\left<\psi(W)u_{\lambda},u_{\lambda}\right>\
\text{ and }\ |\psi(\lambda)|\leq\|\psi(W)\|,$

where $u_{\lambda}:=\frac{z_{\lambda}}{\|z_{\lambda}\|}$ and

$z_{\lambda}:=\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}\frac{\bar{\lambda}_{\alpha}}{{\boldsymbol{\mu}(\alpha,g_{0})}}e_{\alpha}.$

###### Proof.

First, we prove that if
$\lambda=(\lambda_{1},\ldots,\lambda_{n})\in{\mathcal{D}}(\boldsymbol{\mu})$,
then the map $\Phi_{\lambda}:F^{\infty}(\boldsymbol{\mu})\to{\mathbb{C}}$
defined by

$\Phi_{\lambda}(\psi(W)):=\psi(\lambda),\qquad\psi(W)\in
F^{\infty}(\boldsymbol{\mu}),$

is a $w^{*}$-continuous multiplicative linear functional, and

$\Phi_{\lambda}(\psi(W))=\left<\psi(W)1,z_{\lambda}\right>,\qquad\psi(W_{1},\ldots,W_{n})\in
F^{\infty}(\boldsymbol{\mu}).$

Indeed, if $\psi(W):=\sum_{\beta\in{\mathbb{F}}_{n}^{+}}c_{\beta}W_{\beta}$ is
in the Hardy algebra $F^{\infty}(\boldsymbol{\mu})$, then, due to Theorem 4.7
and the identification of multiplier algebra
${\mathcal{M}}^{l}(F^{2}(\boldsymbol{\mu}))$ with
$F^{\infty}(\boldsymbol{\mu})$, we have
$\sum_{\beta\in{\mathbb{F}}_{n}^{+}}|c_{\beta}||\lambda_{\beta}|<\infty$ and

$\begin{split}\left<\psi(W)1,z_{\lambda}\right>&=\left<\sum_{\beta\in{\mathbb{F}}_{n}^{+}}c_{\beta}\boldsymbol{\mu}(\beta,g_{0})e_{\beta},\sum_{\beta\in{\mathbb{F}}_{n}^{+}}\frac{1}{\boldsymbol{\mu}(\beta,g_{0})}\overline{\lambda}_{\beta}e_{\beta}\right>\\\
&=\sum_{\beta\in{\mathbb{F}}_{n}^{+}}c_{\beta}\lambda_{\beta}=\psi(\lambda).\end{split}$

Hence, $\Phi_{\lambda}$ is a $w^{*}$-continuous multiplicative linear
functional. Moreover, for each $\beta\in{\mathbb{F}}_{n}^{+}$, we have

$\begin{split}\left<\psi(W)^{*}z_{\lambda},\boldsymbol{\mu}(\beta,g_{0})e_{\beta}\right>&=\left<z_{\lambda},\psi(W)W_{\beta}(1)\right>\\\
&=\overline{\lambda_{\beta}\psi(\lambda)}=\left<\overline{\psi(\lambda)}z_{\lambda},\boldsymbol{\mu}(\beta,g_{0})e_{\beta}\right>.\end{split}$

Hence, we deduce that
$\psi(W)^{*}z_{\lambda}=\overline{\varphi(\lambda)}z_{\lambda}$ and,
consequently,

$\begin{split}\left<\psi(W)u_{\lambda},u_{\lambda}\right>&=\frac{1}{\|z_{\lambda}\|^{2}}\left<z_{\lambda},\psi(W)^{*}z_{\lambda}\right>\\\
&=\frac{1}{\|z_{\lambda}\|^{2}}\left<z_{\lambda},\overline{\psi(\lambda)}z_{\lambda}\right>=\psi(\lambda).\end{split}$

Now, let $\Phi:F^{\infty}(\boldsymbol{\mu})\to{\mathbb{C}}$ be a
$w^{*}$-continuous multiplicative linear functional. Note that $J:=\ker\Phi$
is a $w^{*}$-closed two-sided ideal of $F^{\infty}(\boldsymbol{\mu})$ of
codimension one. The subspace ${\mathcal{M}}_{J}:=\overline{J(1)}\subset
F^{2}(H_{n})$ has codimension one and
${\mathcal{M}}_{J}+{\mathbb{C}}1=F^{2}(H_{n})$. Indeed, assume that there
exists a vector $y\in({\mathcal{M}}_{J}+{\mathbb{C}}1)^{\perp}$, $\|y\|=1$.
Then we can choose a polynomial $p:=p(W_{1},\ldots,W_{n})\in
F^{\infty}(\boldsymbol{\mu})$ such that $\|p(1)-y\|<1$. Since
$p-\varphi(p)I\in\ker\varphi=J$, we have
$p(1)-\varphi(p)\in{\mathcal{M}}_{J}$. Since
$y\perp{\mathcal{M}}_{J}+{\mathbb{C}}1$, we deduce that

$\begin{split}1&=\|y\|=\left<y-\varphi(p),y\right>\leq|\left<h-p(1),y\right>|+|\left<p(1)-\varphi(p),y\right>|\\\
&=|\left<h-p(1),y\right>|<1,\end{split}$

which is a contradiction. Therefore, $M_{J}$ is an invariant subspace under
$W_{1},\ldots,W_{n}$ and has codimension one. Due to Theorem 4.9, there exists
$\lambda\in{\mathcal{D}}(\boldsymbol{\mu})$ such that
${\mathcal{M}}_{J}=\\{z_{\lambda}\\}^{\perp}$.

Hence, $J:=\ker\Phi\subseteq\ker\Phi_{\lambda}\subset
F^{\infty}(\boldsymbol{\mu})$. Since both $\ker\Phi$ and $\ker\Phi_{\lambda}$
are $w^{*}$-closed two-sided maximal ideals of $F^{\infty}(\boldsymbol{\mu})$
of codimension one, we must have $\ker\Phi=\ker\Phi_{\lambda}$. Therefore,
$\Phi=\Phi_{\lambda}$ and the proof is complete. ∎

##  5\. Functional calculus for arbitrary $n$-tuples of operators

Let $\boldsymbol{\mu}=\\{\mu_{\beta}\\}_{\beta\in{\mathbb{F}}_{n}^{+}}$ be a
sequence of strictly positive real numbers satisfying the boundedness
conditions (4.1) and (4.2) and let $F^{\infty}(\boldsymbol{\mu})$ be the
noncommutative Hardy algebra associated. Define the noncommutative set

${\mathcal{D}}_{\boldsymbol{\mu}}({\mathcal{H}}):=\left\\{(X_{1},\ldots,X_{n})\in
B({\mathcal{H}})^{n}:\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}\frac{1}{\boldsymbol{\mu}(\alpha,g_{0})^{2}}X_{\alpha}X_{\alpha}^{*}<\infty\right\\}.$

We remark that if

$\limsup_{k\to\infty}\left(\sum_{\alpha\in{\mathbb{F}}_{n}^{+},|\alpha|=k}\frac{1}{{\boldsymbol{\mu}(\alpha,g_{0})^{4}}}\right)^{1/2k}<\infty,$

then the noncommutative Reinhardt set
${\mathcal{D}}_{\boldsymbol{\mu}}({\mathcal{H}})$ contains a ball centered at
the origin in $B({\mathcal{H}})^{n}$. Indeed, according to [45], the condition
above implies that the formal series
$\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}\frac{1}{{\boldsymbol{\mu}(\alpha,g_{0})^{2}}}Z_{\alpha}$
is a free holomorphic function on a neighborhood of the origin in
$B({\mathcal{H}})^{n}$. Consequently, if $\rho$ denotes the reciprocal of the
$\limsup$ above and $0<t_{0}<\frac{\rho}{\sqrt{n}}$, then
$(t_{0}I_{\mathcal{H}},\ldots
t_{0}I_{\mathcal{H}})\in[B({\mathcal{H}})^{n}]_{\rho}$ and
$\sum_{k=0}^{\infty}\sum_{|\alpha|=k}a_{\alpha}t_{0}^{k}<\infty$. Take
$0<\epsilon<\sqrt{t_{0}}$ and assume that
$X=(X_{1},\ldots,X_{n})\in[B({\mathcal{H}})^{n}]_{\epsilon}$. Then we have

$\sum_{k=0}^{\infty}\sum_{|\alpha|=k}\frac{1}{{\boldsymbol{\mu}(\alpha,g_{0})^{2}}}\|X_{\alpha}X_{\alpha}^{*}\|\leq\sum_{k=0}^{\infty}\sum_{|\alpha|=k}\frac{1}{{\boldsymbol{\mu}(\alpha,g_{0})^{2}}}\epsilon^{2}k\leq\sum_{k=0}^{\infty}\sum_{|\alpha|=k}a_{\alpha}t_{0}^{k}<\infty,$

which proves our assertion.

In what follows, we present a $w^{*}$-continuous
$F^{\infty}(\boldsymbol{\mu})$-functional calculus for the elements in the
noncommutative Reinhardt set
${\mathcal{D}}_{\boldsymbol{\mu}}({\mathcal{H}})$, where ${\mathcal{H}}$ is a
separable Hilbert space.

######  Theorem 5.1.

Let $T=(T_{1},\ldots,T_{n})\in{\mathcal{D}}_{\boldsymbol{\mu}}({\mathcal{H}})$
and let $\Psi_{T}:F^{\infty}(\boldsymbol{\mu})\to B({\mathcal{H}})$ be defined
by

$\Psi_{T}(\varphi(W))=\varphi(T)=:\text{\rm
SOT-}\lim_{N\to\infty}\sum_{|\alpha|\leq
N}\left(1-\frac{|\alpha|}{N+1}\right)c_{\alpha}T_{\alpha},$

where $\varphi(W)\in F^{\infty}(\boldsymbol{\mu})$ has the Fourier
representation $\sum_{\alpha\in{\mathbb{F}}_{n}}c_{\alpha}W_{\alpha}$. Then
$\Psi_{T}$ has the following properties.

  1. (i)

$\Psi_{T}\left(\sum_{|\alpha|\leq
m}c_{\alpha}W_{\alpha}\right)=\sum_{|\alpha|\leq m}c_{\alpha}T_{\alpha}$,
$m\in{\mathbb{N}}.$

  2. (ii)

$\Psi_{T}$ is sequentially WOT-(resp. SOT-) continuous.

  3. (iii)

$\Psi_{T}$ is a completely bounded algebra homomorphism and

$\|\Psi_{T}\|_{cb}\leq\left\|\sum_{k=0}^{\infty}\sum_{\alpha\in{\mathbb{F}}_{n}^{+},|\alpha|=k}\frac{1}{\boldsymbol{\mu}(\alpha,g_{0})^{2}}T_{\alpha}T_{\alpha}^{*}\right\|^{1/2}.$

  4. (iv)

$\Psi_{T}$ is $w^{*}$-continuous.

  5. (v)

$r(\varphi(T))\leq r(\varphi(W))$ for any $\varphi(W)\in
F^{\infty}(\boldsymbol{\mu})$, where $r(X)$ denotes the spectral radius of
$X$.

###### Proof.

According to to Corollary 2.2, if
$T=(T_{1},\ldots,T_{n})\in{\mathcal{D}}_{\boldsymbol{\mu}}({\mathcal{H}})$,
then there is a joint invariant subspace ${\mathcal{M}}\subset
F^{2}(H_{n})\otimes{\mathcal{H}}$ under the operators
$W_{1}^{*},\ldots,W_{n}^{*}$ such that

(5.1) $T_{i}^{*}=X^{-1}(W_{i}^{*}\otimes I)|_{\mathcal{M}}X,\qquad
i\in\\{1,\ldots,n\\}.$

Define the map $\Phi_{T}:F^{\infty}(\boldsymbol{\mu})\to B({\mathcal{H}})$ by
setting

(5.2) $\Phi_{T}(\varphi(W)):=X^{*}P_{\mathcal{M}}(\varphi(W)\otimes
I_{\mathcal{H}})|_{\mathcal{M}}(X^{*})^{-1},\qquad\varphi(W)\in
F^{\infty}(\boldsymbol{\mu}).$

Note that relation (5.1) implies
$T_{\alpha}=X^{*}P_{\mathcal{M}}(W_{\alpha}\otimes
I_{\mathcal{H}})|_{\mathcal{M}}(X^{*})^{-1}$ for any
$\alpha\in{\mathbb{F}}_{n}^{+}$. This implies
$\Psi_{T}\left(\sum_{|\alpha|\leq
m}c_{\alpha}W_{\alpha}\right)=\sum_{|\alpha|\leq m}c_{\alpha}T_{\alpha}$ for
any $m\in{\mathbb{N}}$.

Now, we prove that $\Phi_{T}$ is sequentially WOT-(resp. SOT-) continuous. Let
$\\{\zeta_{k}(W)\\}_{k}\subset F^{\infty}(\boldsymbol{\mu})$ be a sequence
such that WOT-$\lim_{k\to\infty}\zeta_{k}(W)=\varphi(W)$. Since the WOT and
$w^{*}$ topologies concide on bounded set, we have
$w^{*}$-$\lim_{k\to\infty}\zeta_{k}(W)=\varphi(W)$ which implies
$w^{*}$-$\lim_{k\to\infty}(\zeta_{k}(W)\otimes
I_{\mathcal{H}})=\varphi(W)\otimes I_{\mathcal{H}}$. Hence,
WOT-$\lim_{k\to\infty}(\zeta_{k}(W)\otimes I_{\mathcal{H}})=\varphi(W)\otimes
I_{\mathcal{H}}$ and
WOT-$\lim_{k\to\infty}\Phi_{T}(\zeta_{k}(W))=\Phi_{T}(\varphi(W))$. Similarly,
if SOT-$\lim_{k\to\infty}\zeta_{k}(W)=\varphi(W)$, then $\|\zeta_{k}(W)\|\leq
M$, $k\in{\mathbb{N}}$, for some $M>0$. Consequently, we have
SOT-$\lim_{k\to\infty}(\zeta_{k}(W)\otimes I_{\mathcal{K}})=\varphi(W)\otimes
I_{\mathcal{H}}$ which implies that
SOT-$\lim_{k\to\infty}\Phi_{T}(\zeta_{k}(W))=\Phi_{T}(\varphi(W))$. This
proves our assertion.

Let $\\{\varphi_{\iota}(W)\\}_{\iota}$ be a net in
$F^{\infty}(\boldsymbol{\mu})$ such that
$w^{*}$-$\lim_{\iota}\varphi_{\iota}(W)=\varphi(W)\in
F^{\infty}(\boldsymbol{\mu})$. Then we deduce that
$w^{*}$-$\lim_{\iota}\varphi_{\iota}(W)\otimes
I_{\mathcal{H}}=\varphi(W)\otimes I_{\mathcal{H}}$ and, consequently,

$w^{*}\text{-}\lim_{\iota}X^{*}P_{\mathcal{M}}(\varphi_{\iota}(W)\otimes
I_{\mathcal{H}})|_{\mathcal{M}}(X^{*})^{-1}=X^{*}P_{\mathcal{M}}(\varphi(W)\otimes
I_{\mathcal{H}})|_{\mathcal{M}}(X^{*})^{-1}.$

This proves that
$w^{*}$-$\lim_{\iota}\Phi_{T}(\varphi_{\iota}(W))=\Phi_{T}(\varphi(W))$.
Therefore, $\Phi_{T}$ is $w^{*}$-continuous.

Let $[\varphi_{i,j}(W)]_{s\times s}$ be an $s\times s$ matrix with entries in
the Hardy algebra $F^{\infty}(\boldsymbol{\mu})$. Note that relation (5.2)
implies

$[\Phi_{T}(\varphi_{ij}(W))]_{s\times
s}:=(\oplus_{1}^{s}X^{*}P_{\mathcal{M}})[\Phi_{T}(\varphi_{ij}(W))]_{s\times
s}\otimes I_{\mathcal{H}})|_{\mathcal{M}}]_{s\times
s}(\oplus_{1}^{s}(X^{*})^{-1}).$

Hence,

$\|[\Phi_{T}(\varphi_{ij}(W))]_{s\times
s}\|\leq\|X\|\|X^{-1}\|[\Phi_{T}(\varphi_{ij}(W))]_{s\times s}\|.$

Therefore, $\Phi_{T}$ is a completely bounded linear map and
$\|\Phi_{T}\|_{cb}\leq\|X\|\|X^{-1}\|$ . Using relations (5.1) and (5.2), it
is easy to see that $\Phi_{T}$ is a homomorphism on the polynomials in
$W_{1},\ldots,W_{n}$. Since these polynomials are $w^{*}$-dense in
$F^{\infty}(\boldsymbol{\mu})$ and $\Phi_{T}$ is $w^{*}$-continuous, an
approximation argument shows that $\Phi_{T}$ is a homomorphism on
$F^{\infty}(\boldsymbol{\mu})$.

Now, let $\varphi(W)\in F^{\infty}(\boldsymbol{\mu})$ have the Fourier
representation $\sum_{\alpha\in{\mathbb{F}}_{n}}c_{\alpha}W_{\alpha}$ and
define

$p_{N}(W):=\sum_{|\alpha|\leq
N}\left(1-\frac{|\alpha|}{N+1}\right)c_{\alpha}W_{\alpha}.$

According to Theorem 4.4, $\text{\rm
SOT-}\lim_{N\to\infty}p_{N}(W)=\varphi(W)$ and
$\|p_{N}(W)\|\leq\|\varphi(W)\|$ for any $N\in{\mathbb{N}}$. Then $\text{\rm
SOT-}\lim_{N\to\infty}(p_{N}(W)\otimes I_{\mathcal{H}})=\varphi(W)\otimes
I_{\mathcal{H}}$. On the other hand, due to relation (5.2), we have

$p_{N}(T)=X^{*}P_{\mathcal{M}}(p_{N}(W)\otimes
I_{\mathcal{H}})|_{\mathcal{M}}(X^{*})^{-1}.$

Hence, we deduce that

$\text{\rm
SOT-}\lim_{N\to\infty}p_{N}(T)=X^{*}P_{\mathcal{M}}(\varphi(W)\otimes
I_{\mathcal{H}})|_{\mathcal{M}}(X^{*})^{-1}=\Phi_{T}(\varphi(W)).$

Therefore, $\Psi_{T}(\varphi(W))=\Phi_{T}(\varphi(W))$ for any $\varphi(W)\in
F^{\infty}(\boldsymbol{\mu})$.

It remains to prove part(v). To this end, note that due to the properties of
the map $\Psi_{T}$, we have

$\varphi(T)^{k}=X^{*}P_{\mathcal{M}}(\varphi(W)^{k}\otimes
I_{\mathcal{H}})|_{\mathcal{M}}(X^{*})^{-1},\qquad k\in{\mathbb{N}}.$

Hence we deduce that
$\|\varphi(T)^{k}\|^{1/k}\leq\|X\|^{1/k}\|X^{-1}\|^{1/k}\|\varphi(W)^{k}\|^{1/k}$.
Passing to the limit as $k\to\infty$, we conclude that $r(\varphi(T))\leq
r(\varphi(W))$. The proof is complete. ∎

######  Theorem 5.2.

If $T=(T_{1},\ldots,T_{n})\in B({\mathcal{H}})^{n}$, then there exist a
weighted multi-shift $W=(W_{1},\ldots,W_{n})$ associated with some weights
$\boldsymbol{\mu}=\\{\mu_{\beta}\\}_{\beta\in{\mathbb{F}}_{n}^{+}}$ satisfying
the conditions (4.1), (4.2), and $r(W)=r(T)$, and such that the map
$\Psi_{T}:F^{\infty}(\boldsymbol{\mu})\to B({\mathcal{H}})$ defined by

$\Psi_{T}(\varphi(W))=\varphi(T)=:\text{\rm
SOT-}\lim_{N\to\infty}\sum_{|\alpha|\leq
N}\left(1-\frac{|\alpha|}{N+1}\right)c_{\alpha}T_{\alpha},$

where $\varphi(W)\in F^{\infty}(\boldsymbol{\mu})$ has the Fourier
representation $\sum_{\alpha\in{\mathbb{F}}_{n}}c_{\alpha}W_{\alpha}$, has all
the properties listed in Theorem 5.1 and
$\|\Psi_{T}\|_{cb}\leq\frac{\pi}{\sqrt{6}}$. When $T=(T_{1},\ldots,T_{n})$ is
a nilpotent $n$-tuple of order $m\geq 2$, we have

$\|\Psi_{T}\|_{cb}\leq\sqrt{\sum_{k=0}^{m-1}\frac{1}{(k+1)^{2}}}.$

###### Proof.

Assume that $T=(T_{1},\ldots,T_{n})$ is not a nilpotent $n$-tuple of
operators. As in the proof of Theorem 2.5, we define

$\mu_{\beta}:=\frac{|\beta|+1}{|\beta|}\frac{\left\|\sum\limits_{\sigma\in{\mathbb{F}}_{n}^{+},|\sigma|=|\beta|}T_{\sigma}T_{\sigma}^{*}\right\|^{1/2}}{\left\|\sum\limits_{\sigma\in{\mathbb{F}}_{n}^{+},|\sigma|=|\beta|-1}T_{\sigma}T_{\sigma}^{*}\right\|^{1/2}},\qquad|\beta|\geq
1.$

In this case, we have

$\begin{split}\boldsymbol{\mu}(\alpha,g_{0})&=(|\alpha|+1)\left\|\sum\limits_{\sigma\in{\mathbb{F}}_{n}^{+},|\sigma|=|\alpha|}T_{\sigma}T_{\sigma}^{*}\right\|^{1/2}\\\
\frac{\boldsymbol{\mu}(\alpha
g_{i},g_{0})}{\boldsymbol{\mu}(\alpha,g_{0})}&=\frac{|\alpha|+2}{|\alpha|+1}\cdot\frac{\left\|\sum\limits_{\sigma\in{\mathbb{F}}_{n}^{+},|\sigma|=|\alpha|+1}T_{\sigma}T_{\sigma}^{*}\right\|^{1/2}}{\left\|\sum\limits_{\sigma\in{\mathbb{F}}_{n}^{+},|\sigma|=|\alpha|}T_{\sigma}T_{\sigma}^{*}\right\|^{1/2}}\leq\frac{|\alpha|+2}{|\alpha|+1}\left\|\sum_{i=1}^{n}T_{i}T_{i}^{*}\right\|^{1/2}.\end{split}$

Consequently, the conditions (4.1), (4.2) are satisfied. Let
$W=(W_{1},\ldots,W_{n})$ be the weighted multi-shift associated with the
weights $\\{\mu_{\beta}\\}_{\beta\in{\mathbb{F}}_{n}^{+},|\beta|\geq 1}.$ The
fact that $r(W)=r(T)$ is due to Theorem 2.5. On the other hand, Theorem 5.1
shows that the map $\Psi_{T}$ has all the properties listed in the same
theorem. It remains to prove that $\|\Psi_{T}\|_{cb}\leq\frac{\pi}{\sqrt{6}}$.
Note that, according to the proof of Theorem 5.1,
$\|\Psi_{T}\|_{cb}\leq\|X\|\|X^{-1}\|$, where $X$ is the invertible operator
implementing the similarity of $T$ with the compression of $W$ to the
appropriate subspace. In our case, we have $X=K_{\boldsymbol{\mu}}$ and
$\|K_{\boldsymbol{\mu}}\|\|K_{\boldsymbol{\mu}}^{-1}\|\leq\frac{\pi}{\sqrt{6}}$
(see the proof of Theorem 2.5). The case when $T=(T_{1},\ldots,T_{n})$ is a
nilpotent $n$-tuple of operators can be treated in a similar manner. The proof
is complete ∎

######  Corollary 5.3.

In the setting of Theorem 5.2, the joint point spectrum
$\sigma_{p}(W_{1}^{*},\ldots,W_{n}^{*})$ satisfies the following relations

$({\mathbb{C}}^{n})_{r(T)}\subset\sigma_{p}(W_{1}^{*},\ldots,W_{n}^{*})\subset\overline{({\mathbb{C}}^{n})}_{r(W)}.$

###### Proof.

According to Corollary 4.9, the joint point spectrum
$\sigma_{p}(W_{1}^{*},\ldots,W_{n}^{*})$ coincides with the set

$\\{\lambda=(\lambda_{1},\ldots,\lambda_{n})\in{\mathbb{C}}^{n}:\
\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}\frac{|\lambda_{\alpha}|^{2}}{{\boldsymbol{\mu}(\alpha,g_{0})^{2}}}<\infty\\}$

and
$\sigma_{p}(W_{1}^{*},\ldots,W_{n}^{*})\subset\overline{({\mathbb{C}}^{n})}_{r(W)}$.
Analysing the convergence of the series

$\sum_{k=0}^{\infty}\frac{1}{(k+1)^{2}\left\|\sum\limits_{\sigma\in{\mathbb{F}}_{n}^{+},|\sigma|=|\alpha|}T_{\sigma}T_{\sigma}^{*}\right\|}(|\lambda_{1}|^{2}+\cdots+|\lambda_{n}|^{2})^{k}$

and taking into account that
$r(T)=\lim_{k\to\infty}\left\|\sum\limits_{\sigma\in{\mathbb{F}}_{n}^{+},|\sigma|=k}T_{\sigma}T_{\sigma}^{*}\right\|^{1/2k}$,
one can easily deduce the inclusion
$({\mathbb{C}}^{n})_{r(T)}\subset\sigma_{p}(W_{1}^{*},\ldots,W_{n}^{*})$. ∎

In what follows we present a spectral version of the Schwarz lemma for the
noncommutative Hardy algebra $F^{\infty}(\boldsymbol{\mu})$.

######  Theorem 5.4.

Let $\boldsymbol{\mu}=\\{\mu_{\beta}\\}_{\beta\in{\mathbb{F}}_{n}^{+}}$ be a
sequence of strictly positive real numbers with $\mu_{\alpha}\geq M>0$ for any
$\alpha\in{\mathbb{F}}_{n}^{+}$ and satisfying relations (4.1) and (4.2). If
$\varphi(W)\in F^{\infty}(\boldsymbol{\mu})$ has the properties that
$\varphi(0)=0$ and $\|\varphi(W)\|\leq 1$, then

$r(\varphi(X))\leq r(X),\qquad
X\in{\mathcal{D}}_{\boldsymbol{\mu}}({\mathcal{H}}).$

###### Proof.

Let $\varphi(W)\in F^{\infty}(\boldsymbol{\mu})$ have the Fourier
representation $\sum_{\beta\in{\mathbb{F}}_{n}^{+},|\alpha|\geq
1}c_{\beta}W_{\beta}$. Then for any polynomial
$p\in\operatorname{span}\left\\{e_{\alpha}:\
\alpha\in{\mathbb{F}}_{n}^{+}\right\\}$, we have

$\varphi(W)p=\sum_{i=1}^{n}W_{i}\left(\sum_{\gamma\in{\mathbb{F}}_{n}^{+}}c_{g_{i}\gamma}W_{\gamma}p\right)=\sum_{i=1}^{n}W_{i}\Phi_{i}(W)p$

where
$\Phi_{i}(W)p:=\sum_{\gamma\in{\mathbb{F}}_{n}^{+}}c_{g_{i}\gamma}W_{\gamma}p$.
Since the weights $\\{\mu_{\beta}\\}_{\beta\in{\mathbb{F}}_{n}^{+}}$ are
bounded away from zero, we have $W_{i}=S_{i}D_{i}$, where $D_{i}$ is an
invertible positive diagonal operator. Note that

$\|W_{i}^{*}\varphi(W)\|\|p\|\geq\|W_{i}^{*}\varphi(W)p\|=\|D_{i}^{2}\Phi_{i}(W)p\|\geq\frac{1}{\|D_{i}^{-1}\|^{2}}\|\Phi_{i}(W)p\|$

for any $p\in\operatorname{span}\left\\{e_{\alpha}:\
\alpha\in{\mathbb{F}}_{n}^{+}\right\\}$. Hence, we deduce that $\Phi_{i}(W)\in
F^{\infty}(\boldsymbol{\mu})$, $i\in\\{1,\ldots,n\\}$, and we have the Gleason
type decomposition

(5.3) $\varphi(W)=\sum_{i=1}^{n}W_{i}\Phi_{i}(W).$

Taking into account that the operators $W_{1},\ldots,W_{n}$ have orthogonal
ranges, we deduce that

$\|\varphi(W)^{*}\varphi(W)\|=\|\sum_{i=1}^{n}\Phi_{i}(W)^{*}D_{i}^{2}\Phi_{i}(W)\|\geq
M\|\sum_{i=1}^{n}\Phi_{i}(W)^{*}\Phi_{i}(W)\|$

for some constant $M>0$. Due to Theorem 5.1, relation (5.3) implies
$\varphi(X)=\sum_{i=1}^{n}X_{i}\Phi_{i}(X)$ for any
$X\in{\mathcal{D}}_{\boldsymbol{\mu}}({\mathcal{H}})$. The same theorem shows
that the map $\Psi_{T}$ is completely bounded, which show that

$\left\|\left[\begin{matrix}\Phi_{1}(X)\\\ \vdots\\\
\Phi_{n}(X)\end{matrix}\right]\right\|\leq
K\left\|\left[\begin{matrix}\Phi_{1}(W)\\\ \vdots\\\
\Phi_{n}(W)\end{matrix}\right]\right\|$

for some constant $K>0$. Using this inequality, we obtain

$\begin{split}\|\varphi(X)\|&\leq\|[X_{1}\cdots
X_{n}]\|\left\|\left[\begin{matrix}\Phi_{1}(X)\\\ \vdots\\\
\Phi_{n}(X)\end{matrix}\right]\right\|\leq K\|[X_{1}\cdots
X_{n}]\|\left\|\left[\begin{matrix}\Phi_{1}(W)\\\ \vdots\\\
\Phi_{n}(W)\end{matrix}\right]\right\|\\\ &=K\|[X_{1}\cdots
X_{n}]\|\|\sum_{i=1}^{n}\Phi_{i}(W)^{*}\Phi_{i}(W)\|^{1/2}\\\
&\leq\frac{K}{\sqrt{M}}\|[X_{1}\cdots
X_{n}]\|\|\varphi(W)^{*}\varphi(W)\|^{1/2}\\\
&\leq\frac{K}{\sqrt{M}}\|[X_{1}\cdots X_{n}]\|\end{split}$

In a similar manner, we can show that

$\|\varphi(X)^{k}\|\leq C\|[X_{\alpha}:\ |\alpha|=k]||$

for some constant $C>0$ which depends only on $\boldsymbol{\mu}$ and $X$.
Hence, we deduce that

$\ \|\varphi(X)^{k}\|^{1/k}\leq
C^{1/k}\left\|\sum_{|\alpha|=k}X_{\alpha}X_{\alpha}^{*}\right\|^{1/2k},\qquad
k\in{\mathbb{N}}.$

Taking $k\to\infty$, we obtain $r(\varphi(X))\leq r(X)$ and complete the
proof. ∎

######  Theorem 5.5.

Let $W=(W_{1},\ldots,W_{n})$ be the weighted left multi-shift associated with
$\boldsymbol{\mu}=\\{\mu_{\beta}\\}_{|\beta|\geq 1}$, where $\mu_{\beta}>0$
and conditions (4.1), (4.2) are satisfied. If $W$ is a row power bounded with
bound $M>0$, then the map $\Phi_{W}:{\mathcal{A}}_{n}\to B(F^{2}(H_{n}))$
defined by $\Phi_{W}(p(S_{1},\ldots,S_{n})):=p(W_{1},\ldots,W_{n})$ can be
extended to a $w^{*}$-continuous completely bounded map
$\Phi_{W}:F^{\infty}(H_{n})\to F^{\infty}(\boldsymbol{\mu})$ with
$\|\Phi_{W}\|_{cb}\leq M$ such that

$\Phi_{W}(\varphi(S_{1},\ldots,S_{n}))=\varphi(W_{1},\ldots,W_{n})$

for any $\varphi(S_{1},\ldots,S_{n})\in F^{\infty}(H_{n})$.

###### Proof.

Since $W$ is a row power bounded multi-shift with bound $M>0$, we can apply
Theorem 1.12 and deduce that $W$ is jointly similar to a pure row contraction
and the opearator $A$ implementing the similarity can be chosen such that
$\|A\|\|A^{-1}\|\leq M$. Consequently,
$p(W_{1},\ldots,W_{n})=Ap(S_{1},\ldots,S_{n})A^{-1}$ for any noncommutative
polynomial $p$. Now, using the $w^{*}$-continuous
$F^{\infty}(H_{n})$-functional calculus for pure row contractions, one can
easily complete the proof. ∎

######  Corollary 5.6.

Let $\boldsymbol{\mu}=\\{\mu_{\beta}\\}_{|\beta|\geq 1}$ be a sequence of
strictly positive weights such that the associated the weighted left multi-
shift $W=(W_{1},\ldots,W_{n})$ is row power bounded with bound $M>0$. If
$T=(T_{1},\ldots,T_{n})\in{\mathcal{D}}_{\boldsymbol{\mu}}({\mathcal{H}})$,
then the map $\Gamma_{T}:F^{\infty}(H_{n})\to B({\mathcal{H}})$ defined by

$\Gamma_{T}(\varphi(S))=\varphi(T)=:\text{\rm
SOT-}\lim_{N\to\infty}\sum_{|\alpha|\leq
N}\left(1-\frac{|\alpha|}{N+1}\right)c_{\alpha}T_{\alpha},$

where $\varphi(S)$ has the Fourier representation
$\sum_{\alpha\in{\mathbb{F}}_{n}}c_{\alpha}S_{\alpha}$, has the following
properties.

  1. (i)

$\Gamma_{T}\left(\sum_{|\alpha|\leq
m}c_{\alpha}S_{\alpha}\right)=\sum_{|\alpha|\leq m}c_{\alpha}T_{\alpha},\qquad
m\in{\mathbb{N}}.$

  2. (ii)

$\Gamma_{T}$ is sequentially WOT-(resp. SOT-)continuous.

  3. (iii)

$\Gamma_{T}$ is a completely bounded algebra homomorphism and

$\|\Gamma_{T}\|_{cb}\leq
M\left\|\sum_{k=0}^{\infty}\sum_{\alpha\in{\mathbb{F}}_{n}^{+},|\alpha|=k}\frac{1}{\boldsymbol{\mu}(\alpha,g_{0})^{2}}T_{\alpha}T_{\alpha}^{*}\right\|^{1/2}.$

  4. (iv)

$\Gamma_{T}$ is $w^{*}$-continuous.

  5. (v)

$r(\varphi(T))\leq r(\varphi(S))$ for any $\varphi\in F^{\infty}(H_{n})$,
where $r(X)$ denotes the spectral radius of $X$.

According to [45], a formal power series
$\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}c_{\alpha}Z_{\alpha}$ is a free
holomorphic function in a neighborhood of the origin if there is $r>0$ such
that the series $\sum_{k=0}^{\infty}\sum_{|\alpha|=k}c_{\alpha}X_{\alpha}$ is
convergent in norm for any Hilbert space ${\mathcal{H}}$ and any $n$-tuple
$(X_{1},\ldots,X_{n})\in B({\mathcal{H}})^{n}$ with
$\|X_{1}X_{1}^{*}+\cdots+X_{n}X_{n}^{*}\|<r^{2}$. This condition is equivalent
to the relation

(5.4)
$\limsup_{k\to\infty}\left(\sum_{|\alpha|=k}|c_{\alpha}|^{2}\right)^{1/2k}<\infty.$

We denote by ${\rm Hol}_{0}(Z)$ the algebra of all free holomorphic function
$f$ satisfying the condition above.

######  Theorem 5.7.

Let $T=(T_{1},\ldots,T_{n})\in B({\mathcal{H}})^{n}$ be a quasi-nilpotent
$n$-tuple of operators. Then there exists a quasi-nilpotent weighted multi-
shift $W=(W_{1},\ldots,W_{n})$ with the following properties.

  1. (i)

$\|f(T_{1},\ldots,T_{n})\|\leq\frac{\pi}{\sqrt{6}}\|f(W_{1},\ldots,W_{n})\|$
for any $f\in{\rm Hol}_{0}(Z)$.

  2. (ii)

$r(f(T_{1},\ldots,T_{n}))\leq r(f(W_{1},\ldots,W_{n}))$ for any $f\in{\rm
Hol}_{0}(Z)$.

  3. (iii)

Let $\\{f_{k}\\},f\in{\rm Hol}_{0}(Z)$ be such that
$\\{\|f_{k}(W_{1},\ldots,W_{n})\|\\}_{k}$ is a bounded sequence and

$f_{k}(W_{1},\ldots,W_{n})\to f(W_{1},\ldots,W_{n}),\qquad\text{as }\
k\to\infty,$

in the operator norm (resp. WOT-, $w^{*}$, SOT-) topology, then
$f_{k}(T_{1},\ldots,T_{n})\to f(T_{1},\ldots,T_{n})$ in the corresponding
topology, respectively.

In particular, if $f(S_{1},\ldots,S_{n})\in F^{\infty}(H_{n})$ then

$\|f(T_{1},\ldots,T_{n})\|\leq\frac{\pi}{\sqrt{6}}M\|f(S_{1},\ldots,S_{n})\|,$

where $M$ is the power bound of $W=(W_{1},\ldots,W_{n})$.

###### Proof.

According to Theorem 4.1 from [45], if
$f=\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}c_{\alpha}Z_{\alpha}$ is a free
holomorphic function on a neighborhood of zero, then, for any $n$-tuple
$T=(T_{1},\ldots,T_{n})$ with $r(T)=0$, the series

$f(T_{1},\ldots,T_{n}):=\sum_{k=0}^{\infty}\sum_{|\alpha|=k}c_{\alpha}T_{\alpha}$

is convergent in norm. Applying Theorem 2.5 we find a weighted multi-shift
$W=(W_{1},\ldots,W_{n})$ with $r(W)=r(T)=0$ and such

$T_{i}=X^{*}P_{\mathcal{M}}(W_{i}\otimes I)|_{\mathcal{M}}(X^{*})^{-1},\qquad
i\in\\{1,\ldots,n\\},$

where $X$ is an invertible operator with $\|X\|\leq\frac{\pi}{\sqrt{6}}$. Now,
it is easy to see

$f(T_{1},\ldots,T_{n})=X^{*}P_{\mathcal{M}}(f(W_{1},\ldots,W_{n})\otimes
I)|_{\mathcal{M}}(X^{*})^{-1}.$

As in the the proof of Theorem 5.1 and Theorem 5.2, one can prove items (i),
(ii), and (ii). To prove the last part of the theorem, note that $W$ is power
bounded and, consequently, one can apply Theorem 5.5 and the first part of
this theorem to complete the proof. ∎

##  6\. Function theory on Reinhardt domains and multiplier algebras

Let $\boldsymbol{\mu}=\\{\mu_{\beta}\\}_{|\beta|\geq 1}$ be a weight sequence,
where $\mu_{\beta}>0$ and conditions (4.1), (4.2) are satisfied. For each
$\lambda=(\lambda_{1},\ldots,\lambda_{n})$ and each $n$-tuple ${\bf
k}:=(k_{1},\ldots,k_{n})\in{\mathbb{N}}_{0}^{n}$, where
${\mathbb{N}}_{0}:=\\{0,1,\ldots\\}$, let $\lambda^{\bf
k}:=\lambda_{1}^{k_{1}}\cdots\lambda_{n}^{k_{n}}$. For each ${\bf
k}\in{\mathbb{N}}_{0}$, we denote

$\Lambda_{\bf k}:=\\{\alpha\in{\mathbb{F}}_{n}^{+}:\
\lambda_{\alpha}=\lambda^{\bf k}\text{ for all
}\lambda\in{\mathbb{C}}^{n}\\}.$

If ${\bf k}\in{\mathbb{N}}_{0}^{n}$, we define the vector

$y^{(\bf k)}:=\frac{1}{\omega_{\bf k}}\sum_{\alpha\in\Lambda_{\bf
k}}\frac{1}{\boldsymbol{\mu}(\alpha,g_{0})}e_{\alpha}\in
F^{2}(H_{n}),\quad\text{ where }\ \omega_{\bf k}:=\sum_{\alpha\in\Lambda_{\bf
k}}\frac{1}{\boldsymbol{\mu}(\alpha,g_{0})^{2}}$

and $\boldsymbol{\mu}(\alpha,g_{0})$ is defined in Proposition 1.1. Note that
the set $\\{y^{(\bf k)}:\ {\bf k}\in{\mathbb{N}}_{0}^{n}\\}$ consists of
orthogonal vectors in $F^{2}(H_{n})$ and $\|y^{(\bf
k)}\|=\frac{1}{\sqrt{\omega_{\bf k}}}$. We denote by
$F_{s}^{2}(\boldsymbol{\mu})$ the closed span of these vectors, and call it
the symmetric weighted Fock space associated with the weight sequence
$\boldsymbol{\mu}=\\{\mu_{\beta}\\}_{|\beta|\geq 1}$.

######  Theorem 6.1.

Let $\boldsymbol{\mu}=\\{\mu_{\beta}\\}_{|\beta|\geq 1}$ be a weight sequence
with the property that ${\mathcal{D}}(\boldsymbol{\mu})$ contains a
neighborhood of the origin and let $W=(W_{1}.\ldots,W_{m})$ be the associated
weighted left multi-shift. If $J_{c}$ is the $w^{*}$-closed two-sided ideal of
the Hardy algebra $F^{\infty}(\boldsymbol{\mu})$ generated by the commutators

$W_{i}W_{j}-W_{j}W_{i},\qquad i,j\in\\{1,\ldots,n\\},$

then the following statements hold.

  1. (i)

$F_{s}^{2}(\boldsymbol{\mu})=\overline{\text{\rm span}}\\{z_{\lambda}:\
\lambda\in{\mathcal{D}}(\boldsymbol{\mu})\\}=F^{2}(H_{n})\ominus\overline{J_{c}(1)}$,
where $z_{\lambda}$ is defined in Proposition 4.9.

  2. (ii)

The symmetric weighted Fock space $F_{s}^{2}(\boldsymbol{\mu})$ can be
identified with the Hilbert space
${\mathbb{H}}^{2}({\mathcal{D}}(\boldsymbol{\mu}))$ of all functions
$\varphi:{\mathcal{D}}(\boldsymbol{\mu})\to{\mathbb{C}}$ which admit a power
series representation $\varphi(\lambda)=\sum_{{\bf
k}\in{\mathbb{N}}_{0}}c_{\bf k}\lambda^{\bf k}$ with

$\|\varphi\|_{2}=\sum_{{\bf k}\in{\mathbb{N}}_{0}}|c_{\bf
k}|^{2}\frac{1}{\omega_{\bf k}}<\infty.$

More precisely, every element $\varphi=\sum_{{\bf k}\in{\mathbb{N}}_{0}}c_{\bf
k}y^{(\bf k)}$ in $F_{s}^{2}(\boldsymbol{\mu})$ has a functional
representation on ${\mathcal{D}}(\boldsymbol{\mu})$ given by

$\varphi(\lambda):=\left<\varphi,z_{\lambda}\right>=\sum_{{\bf
k}\in{\mathbb{N}}_{0}}c_{\bf k}\lambda^{\bf
k},\quad\lambda=(\lambda_{1},\ldots,\lambda_{n})\in{\mathcal{D}}(\boldsymbol{\mu}),$

and

$|\varphi(\lambda)|\leq\|\varphi\|_{2}\left(\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}\frac{|\lambda_{\alpha}|^{2}}{\boldsymbol{\mu}(\alpha,g_{0})^{2}}\right)^{1/2},\quad\lambda=(\lambda_{1},\ldots,\lambda_{n})\in{\mathcal{D}}(\boldsymbol{\mu}).$

  3. (iii)

${\mathbb{H}}^{2}({\mathcal{D}}(\boldsymbol{\mu}))$ is the reproducing kernel
Hilbert space with kernel
$\boldsymbol{\kappa}:{\mathcal{D}}(\boldsymbol{\mu})\times{\mathcal{D}}(\boldsymbol{\mu})\to{\mathbb{C}}$
defined by

$\boldsymbol{\kappa}(\zeta,\lambda):=\left<z_{\lambda},z_{\zeta}\right>=\sum_{\beta\in{\mathbb{F}}_{n}^{+}}\frac{1}{\boldsymbol{\mu}(\beta,g_{0})^{2}}\zeta_{\beta}\overline{\lambda}_{\beta}\quad\text{
for all }\ \lambda,\zeta\in{\mathcal{D}}(\boldsymbol{\mu}).$

###### Proof.

First, we note that $\overline{\text{\rm span}}\\{z_{\lambda}:\
\lambda\in{\mathcal{D}}(\boldsymbol{\mu})\\}\subset
F^{2}_{s}(\boldsymbol{\mu})$ due to the fact that

$z_{\lambda}:=\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}\frac{\bar{\lambda}_{\alpha}}{{\boldsymbol{\mu}(\alpha,g_{0})}}e_{\alpha}=\sum_{{\bf
k}\in{\mathbb{N}}_{0}}\bar{\lambda}^{\bf k}\omega_{\bf k}y^{({\bf k})}.$

On the other hand, for any $\gamma,\beta\in{\mathbb{F}}_{n}^{+}$,
$i,j\in\\{1,\ldots,n\\}$, and ${\bf k}\in{\mathbb{N}}_{0}^{n}$, we have

$\begin{split}&\left<y^{({\bf
k})},W_{\gamma}(W_{j}W_{i}-W_{i}W_{j})W_{\beta}1\right>\\\
&\qquad=\frac{1}{\omega_{\bf k}}\left<\sum_{\alpha\in\Lambda_{\bf
k}}\frac{1}{\boldsymbol{\mu}(\alpha,g_{0})}e_{\alpha},\boldsymbol{\mu}(\gamma
g_{j}g_{i}\beta,g_{0})e_{\gamma g_{j}g_{i}\beta}-\boldsymbol{\mu}(\gamma
g_{i}g_{j}\beta,g_{0})e_{\gamma g_{i}g_{j}\beta}\right>=0.\end{split}$

Hence, we deduce that $F_{s}^{2}(\boldsymbol{\mu})\subset
F^{2}(H_{n})\ominus\overline{J_{c}(1)}$. Now, assume that there is a vector
$v=\sum_{\beta\in{\mathbb{F}}_{n}^{+}}c_{\beta}e_{\beta}\in
F^{2}(H_{n})\ominus\overline{J_{c}(1)}$ such that
$\left<v,z_{\lambda}\right>=0$ for any
$\lambda\in{\mathcal{D}}(\boldsymbol{\mu})$. Then we have

$\sum_{{\bf k}\in{\mathbb{N}}_{0}}\left(\sum_{\beta\in\Lambda_{\bf
k}}c_{\beta}\frac{1}{\boldsymbol{\mu}(\beta,g_{0})}\right)\lambda^{\bf
k}=0,\qquad\lambda\in{\mathcal{D}}(\boldsymbol{\mu}).$

Since ${\mathcal{D}}(\boldsymbol{\mu})$ contains a ball centered at the
origin, the uniqueness of the power series representations of holomorphic
functions on domains in ${\mathbb{C}}^{n}$ implies $\sum_{\beta\in\Lambda_{\bf
k}}c_{\beta}\frac{1}{\boldsymbol{\mu}(\beta,g_{0})}=0$ for any ${\bf
k}\in{\mathbb{N}}_{0}^{n}$.

Let $\beta_{0}=\gamma g_{j}g_{i}\omega\in\Lambda_{\bf k}$ and let
$\beta=\gamma g_{i}g_{j}\omega\in\Lambda_{\bf k}$, where
$\gamma,\omega\in{\mathbb{F}}_{n}^{+}$ and $i\neq j$. Since $v\in
F^{2}(H_{n})\ominus\overline{J_{c}(1)}$, we have

$\left<v,W_{\gamma}(W_{j}W_{i}-W_{i}W_{j})W_{\omega}(1)\right>=0,$

which implies
$c_{\beta_{0}}\boldsymbol{\mu}(\beta_{0},g_{0})=c_{\beta}\boldsymbol{\mu}(\beta,g_{0})$.
Since any element $\gamma\in\Lambda_{\bf k}$ can be obtained from $\beta_{0}$
by successive transpositions, repeating the above argument, we deduce that
$c_{\beta_{0}}\boldsymbol{\mu}(\beta_{0},g_{0})=c_{\gamma}\boldsymbol{\mu}(\gamma,g_{0})$
for all $\gamma\in\Lambda_{\bf k}.$ Since $\sum_{\beta\in\Lambda_{\bf
k}}c_{\beta}\frac{1}{\boldsymbol{\mu}(\beta,g_{0})}=0$ for any ${\bf
k}\in{\mathbb{N}}_{0}^{n}$, we conclude that $c_{\beta}=0$ for any
$\beta\in\Lambda_{\bf k}$ and ${\bf k}\in{\mathbb{N}}_{0}^{n}$, so $v=0$.
Consequently, we have $\overline{\text{\rm span}}\\{z_{\lambda}:\
\lambda\in{\mathcal{D}}(\boldsymbol{\mu})\\}=F^{2}(H_{n})\ominus\overline{J_{c}(1)}$,
which completes the proof of part (i).

To prove part (ii), we remark that

$\begin{split}\left<y^{({\bf k})},z_{\lambda}\right>&=\frac{1}{\omega_{\bf
k}}\left<\sum_{\alpha\in\Lambda_{\bf
k}}\frac{1}{\boldsymbol{\mu}(\alpha,g_{0})}e_{\alpha},z_{\lambda}\right>\\\
&=\frac{1}{\omega_{\bf k}}\sum_{\alpha\in\Lambda_{\bf
k}}\frac{1}{\boldsymbol{\mu}(\alpha,g_{0})}\lambda_{\alpha}=\lambda^{\bf
k}\end{split}$

for any ${\bf k}\in{\mathbb{N}}_{0}^{n}$ and
$\lambda\in{\mathcal{D}}(\boldsymbol{\mu})$. This shows that every element
$\psi=\sum_{{\bf k}\in{\mathbb{N}}_{0}}c_{\bf k}y^{({\bf k})}\in
F_{s}^{2}(\boldsymbol{\mu})$ has a functional representation on
${\mathcal{D}}(\boldsymbol{\mu})$ given by
$\psi(\lambda):=\left<\psi,z_{\lambda}\right>=\sum_{{\bf
k}\in{\mathbb{N}}_{0}}c_{\bf k}\lambda^{\bf k}$ and, due to Cauchy-Schwartz
inequality,
$|\psi(\lambda)|\leq\|\psi\|_{2}\left(\sum_{\alpha\in{\mathbb{F}}_{n}^{+}}\frac{|\lambda_{\alpha}|^{2}}{\boldsymbol{\mu}(\alpha,g_{0})^{2}}\right)^{1/2}$.
We remark that any function
$\varphi\in{\mathbb{H}}^{2}({\mathcal{D}}(\boldsymbol{\mu}))$ is uniquely
determined by the coefficients of its power series representation
$\varphi(\lambda)=\sum_{{\bf k}\in{\mathbb{N}}_{0}}c_{\bf k}\lambda^{\bf k}$
due to the uniqueness of the power series representations of holomorphic
functions on domains in ${\mathbb{C}}^{n}$. Now, it is clear that the map
$U:F_{s}^{2}(\boldsymbol{\mu})\to{\mathbb{H}}^{2}({\mathcal{D}}(\boldsymbol{\mu}))$
defined by

$U\left(\sum_{{\bf k}\in{\mathbb{N}}_{0}}c_{\bf k}y^{({\bf
k})}\right):=\sum_{{\bf k}\in{\mathbb{N}}_{0}}c_{\bf k}\lambda^{\bf k}$

is a unitary operator. This completes the proof of part (ii). Note that part
(iii) follows easily from (ii). ∎

We define the operator $B_{i}\in B(F^{2}_{s}(\boldsymbol{\mu}))$ by setting
$B_{i}:=P_{F^{2}_{s}(\boldsymbol{\mu})}W_{i}|_{F^{2}_{s}(\boldsymbol{\mu})}$,
for each $i\in\\{1,\ldots,n\\}$.

######  Theorem 6.2.

Let $\boldsymbol{\mu}=\\{\mu_{\beta}\\}_{|\beta|\geq 1}$ be a weight sequence
with the property that ${\mathcal{D}}(\boldsymbol{\mu})$ contains a
neighborhood of the origin and let $B_{1},\ldots,B_{n}$ be the compressions of
$W_{1},\ldots,W_{n}$ to the symmetric weighted Fock space
$F_{s}^{2}(\boldsymbol{\mu})$, respectively. Then the following statements
hold.

  1. (i)

The $n$-tuple $(B_{1},\ldots,B_{n})$ is unitarily equivalent to
$(M_{\lambda_{1}},\ldots,M_{\lambda_{n}})$, where $M_{\lambda_{i}}$ is the
multiplication on ${\mathbb{H}}^{2}({\mathcal{D}}(\boldsymbol{\mu}))$ by the
coordinate function $\lambda_{i}$.

  2. (ii)

$\sigma_{p}(B_{1}^{*},\ldots,B_{n}^{*})=\sigma_{p}(W_{1}^{*},\ldots,W_{n}^{*})={\mathcal{D}}(\boldsymbol{\mu})$

  3. (iii)

$A\in\\{B_{1},\ldots,B_{n}\\}^{\prime}$ if and only if there is a unique
multiplier $\varphi$ of ${\mathbb{H}}^{2}({\mathcal{D}}(\boldsymbol{\mu}))$
such that

$M_{\varphi}=UAU^{*},$

where the unitary operator
$U:F_{s}^{2}(\boldsymbol{\mu})\to{\mathbb{H}}^{2}({\mathcal{D}}(\boldsymbol{\mu}))$
is defined by $U(y^{({\bf k})}):=\lambda^{\bf k}$.

  4. (iv)

Each $A\in\\{B_{1},\ldots,B_{n}\\}^{\prime}$ is uniquely determined by the
vector

$A1=\sum_{{\bf k}\in{\mathbb{N}}_{0}^{n}}c_{\bf k}y^{({\bf k})}\in
F_{s}^{2}(\boldsymbol{\mu})$

and

$A\zeta=\sum_{{\bf p}\in{\mathbb{N}}_{0}^{n}}\sum_{{\bf
k}\in{\mathbb{N}}_{0}^{n}}c_{\bf k}a_{\bf p}y^{({\bf k}+{\bf p})},\qquad\text{
if }\ \zeta=\sum_{{\bf p}\in{\mathbb{N}}_{0}^{n}}a_{\bf p}y^{({\bf p})}\in
F_{s}^{2}(\boldsymbol{\mu}).$

  5. (v)

$\left<A1,z_{\lambda}\right>=\varphi(\lambda):=\sum_{{\bf
k}\in{\mathbb{N}}_{0}^{n}}c_{\bf k}\lambda^{\bf k}$ and
$A^{*}z_{\lambda}=\overline{\varphi(\lambda)}z_{\lambda}$ for any
$\lambda\in{\mathcal{D}}(\boldsymbol{\mu})$.

###### Proof.

According to the proof of Theorem 4.14 , for any $\varphi(W)\in
F^{\infty}(\boldsymbol{\mu})$ and
$\lambda=(\lambda_{1},\ldots,\lambda_{n})\in{\mathcal{D}}(\boldsymbol{\mu})$,
we have $\varphi(W)^{*}z_{\lambda}=\overline{\varphi(\lambda)}z_{\lambda}$.
Due to the fact that $z_{\lambda}\in F^{2}(\boldsymbol{\mu})$, for any $f\in
F_{s}^{2}(\boldsymbol{\mu})$, we have

$\begin{split}\left<P_{F^{2}_{s}(\boldsymbol{\mu})}\varphi(W)|_{F^{2}_{s}(\boldsymbol{\mu})}f,z_{\lambda}\right>&=\left<f,\varphi(W)^{*}z_{\lambda}\right>\\\
&=\varphi(\lambda)\left<f,z_{\lambda}\right>=\varphi(\lambda)f(\lambda).\end{split}$

Consequently,
$P_{F^{2}_{s}(\boldsymbol{\mu})}\varphi(W)|_{F^{2}_{s}(\boldsymbol{\mu})}$ is
unitarily equivalent to the multiplication operator $M_{\varphi}\in
B({\mathbb{H}}^{2}({\mathcal{D}}(\boldsymbol{\mu})))$. In particular,
$(B_{1},\ldots,B_{n})$ is unitarily equivalent to
$(M_{\lambda_{1}},\ldots,M_{\lambda_{n}})$.

Taking into account that $W_{i}^{*}z_{\lambda}=\bar{\lambda}_{i}z_{\lambda}$
for any
$\lambda=(\lambda_{1},\ldots,\lambda_{n})\in{\mathcal{D}}(\boldsymbol{\mu})$
and, due to Theorem 6.1, $F_{s}^{2}(\boldsymbol{\mu})=\overline{\text{\rm
span}}\\{z_{\lambda}:\ \lambda\in{\mathcal{D}}(\boldsymbol{\mu})\\}$, we
deduce that $W_{i}^{*}F_{s}^{2}(\boldsymbol{\mu})\subset
F_{s}^{2}(\boldsymbol{\mu})$ and
$B_{i}^{*}z_{\lambda}=\bar{\lambda}_{i}z_{\lambda}$. Conversely, assume that
$\lambda=(\lambda_{1},\ldots,\lambda_{n})\in{\mathbb{C}}^{n}$ and there is
$z\in F_{s}^{2}(\boldsymbol{\mu})$, $z\neq 0$, such that
$B_{i}^{*}z=\bar{\lambda}_{i}z$ for any $i\in\\{1,\ldots,n\\}$. Since
$B_{i}^{*}=W_{i}^{*}|_{F_{s}^{2}(\boldsymbol{\mu})}$, we also have
$W_{i}^{*}z=\bar{\lambda}_{i}z$ which, due to Proposition 4.9, implies
$\lambda\in\sigma_{p}(W_{1}^{*},\ldots,W_{n}^{*})$. This proves item (ii).

To prove part (iii), note that, for any ${\bf
k}=(k_{1},\ldots,k_{n})\in{\mathbb{N}}_{0}^{n}$, we have

$\begin{split}\left<B_{i}y^{({\bf k})},z_{\lambda}\right>&=\left<y^{({\bf
k})},W_{i}^{*}z_{\lambda}\right>=\left<y^{({\bf
k})},\bar{\lambda}_{i}z_{\lambda}\right>\\\ &=\lambda_{i}\lambda^{({\bf
k})}=\left<y^{(k_{1},\ldots,k_{j}+1,\ldots
k_{n})},z_{\lambda}\right>\end{split}$

for any $\lambda\in{\mathcal{D}}(\boldsymbol{\mu})$. Hence, $B_{i}y^{({\bf
k})}=y^{(k_{1},\ldots,k_{j}+1,\ldots k_{n})}$ which implies $A(y^{({\bf
k})})=AB^{{\bf k}}1=B^{{\bf k}}A1$. This proves that any
$A\in\\{B_{1},\ldots,B_{n}\\}^{\prime}$ is uniquely determined by $A1\in
F_{s}^{2}(\boldsymbol{\mu})$. Setting
$\varphi:=UA1\in{\mathbb{H}}^{2}({\mathcal{D}}(\boldsymbol{\mu}))$, we
complete the proof of part (iii).

To prove item (iv), assume that $A1=\sum_{{\bf
k}\in{\mathbb{N}}_{0}^{n}}c_{\bf k}y^{({\bf k})}\in
F_{s}^{2}(\boldsymbol{\mu})$. Note that

$A(y^{({\bf p})})=B^{{\bf p}}A1=\sum_{{\bf k}\in{\mathbb{N}}_{0}^{n}}c_{\bf
k}B^{{\bf p}}y^{({\bf k})}=\sum_{{\bf k}\in{\mathbb{N}}_{0}^{n}}c_{\bf
k}y^{({\bf p}+{\bf k})},\qquad{\bf p}\in{\mathbb{N}}_{0}^{n}.$

Now, one can easily deduce part (iv). The first part of item (v) is clear. To
prove the second part, note that

$\begin{split}\left<A^{*}z_{\lambda},y^{({\bf
k})}\right>&=\left<z_{\lambda},Ay^{({\bf k})}\right>=\left<z_{\lambda},B^{{\bf
k}}A1\right>\\\ &=\left<(B^{{\bf
k})^{*}}z_{\lambda},A1\right>=\left<\bar{\lambda}^{\bf
k}z_{\lambda},\sum_{{\bf k}\in{\mathbb{N}}_{0}^{n}}c_{\bf k}y^{({\bf
k})}\right>\\\ &=\left<\overline{\varphi(\lambda)}z_{\lambda},y^{({\bf
k})}\right>\end{split}$

for any ${\bf k}=(k_{1},\ldots,k_{n})\in{\mathbb{N}}_{0}^{n}$. Consequently,
$A^{*}z_{\lambda}=\overline{\varphi(\lambda)}z_{\lambda}$ for any
$\lambda\in{\mathcal{D}}(\boldsymbol{\mu})$. This completes the proof. ∎

Under the notation of the theorem above, we call the formal series $\sum_{{\bf
k}\in{\mathbb{N}}_{0}^{n}}c_{\bf k}B^{\bf k}$ the Fourier representation of
$A$ and note that $\sum_{{\bf k}\in{\mathbb{N}}_{0}^{n}}c_{\bf k}B^{\bf
k}y^{({\bf p})}=Ay^{({\bf p})}$ for any ${\bf p}\in{\mathbb{N}}_{0}^{n}$. This
shows that $A$ is uniquely determined by its Fourier representation.

Similarly to the proof of Theorem 9.2 from [51], one can show that if
$A\in\\{B_{1},\ldots,B_{n}\\}^{\prime}$ has the Fourier representation
$\sum_{{\bf k}\in{\mathbb{N}}_{0}^{n}}c_{\bf k}B^{\bf k}$, then

$A=\text{SOT-}\lim_{N\to\infty}\sum_{{\bf k}\in{\mathbb{N}}_{0}^{n},|{\bf
k}|\leq N}\left(1-\frac{|{\bf k}|}{N+1}\right)c_{\bf k}B^{\bf k},$
$\left\|\sum_{{\bf k}\in{\mathbb{N}}_{0}^{n},|{\bf k}|\leq
N}\left(1-\frac{|{\bf k}|}{N+1}\right)c_{\bf k}B^{\bf
k}\right\|\leq\|A\|,\qquad N\in{\mathbb{N}},$

and

$\left\|\sum_{{\bf k}\in{\mathbb{N}}_{0}^{n},|{\bf k}|=m}c_{\bf k}B^{\bf
k}\right\|\leq\|A\|,\qquad m\in{\mathbb{N}}.$

We introduce the Hardy algebra $F_{s}^{\infty}(\boldsymbol{\mu})$ to be the
$w^{*}$-closed non-self-adjoint algebra generated by the operators
$B_{1},\ldots,B_{n}$ and the identity. Since $B_{i}B_{j}=B_{j}B_{i}$ for any
$i,j\in\\{1,\ldots,n\\}$, one can use the results above to show that

$F_{s}^{\infty}(\boldsymbol{\mu})=\\{B_{1},\ldots,B_{n}\\}^{\prime}=\overline{{\mathcal{P}}(B)}^{SOT}=\overline{{\mathcal{P}}(B)}^{WOT}=\overline{{\mathcal{P}}(B)}^{w^{*}},$

where ${\mathcal{P}}(B)$ stands for the algebra of all polynomials in
$B_{1},\ldots,B_{n}$ and the identity. Moreover,
$F_{s}^{\infty}(\boldsymbol{\mu})$ is the sequential SOT-(resp. WOT-, w*)
closure of ${\mathcal{P}}(B)$.

Using Theorem 6.2 and the above-mentioned results, one can easily prove the
following.

######  Theorem 6.3.

The Hardy algebra $F_{s}^{\infty}(\boldsymbol{\mu})$ can be identified with
the multiplier algebra ${\mathcal{M}}({\mathbb{H}}^{2}(\boldsymbol{\mu}))$.
Moreover, each $\varphi(B)\in F_{s}^{\infty}(\boldsymbol{\mu})$ has a unique
Fourier representation $\sum_{{\bf k}\in{\mathbb{N}}_{0}^{n}}c_{\bf k}B^{\bf
k}$ and

$\varphi(B)=\text{\rm SOT-}\lim_{N\to\infty}\sum_{{\bf
k}\in{\mathbb{N}}_{0}^{n},|{\bf k}|\leq N}\left(1-\frac{|{\bf
k}|}{N+1}\right)c_{\bf k}B^{\bf k}.$

If $A\in B({\mathcal{H}})$ then the set of all invariant subspaces of $A$ is
denoted by $\text{\rm Lat}A$. For any ${\mathcal{U}}\subset B({\mathcal{H}})$
we define $\text{\rm Lat}\,{\mathcal{U}}=\bigcap_{A\in{\mathcal{U}}}\text{\rm
Lat}\,A.$ If ${\mathcal{S}}$ is any collection of subspaces of
${\mathcal{H}}$, then we define $\text{\rm Alg}\,{\mathcal{S}}$ by setting
$\text{\rm Alg}\,{\mathcal{S}}:=\\{A\in B({\mathcal{H}}):\
{\mathcal{S}}\subset\text{\rm Lat}\,A\\}.$ We recall that the algebra
${\mathcal{U}}\subset B({\mathcal{H}})$ is reflexive if
${\mathcal{U}}=\text{\rm Alg Lat}\,{\mathcal{U}}$.

######  Theorem 6.4.

The algebra $F_{s}^{\infty}(\boldsymbol{\mu})$ is reflexive.

###### Proof.

Let $A\in B(F_{s}^{2}(\boldsymbol{\mu}))$ be an operator that leaves invariant
all the invariant subspaces under the operators $B_{1},\ldots,B_{n}$.
According to Theorem 6.2, for any $\lambda\in{\mathcal{D}}(\boldsymbol{\mu})$,
the subspace ${\mathbb{C}}z_{\lambda}$ is invariant under
$B_{1}^{*},\ldots,B_{n}^{*}$. Consequently,
$A^{*}z_{\lambda}=\overline{\varphi(\lambda)}z_{\lambda}$ for some function
$\varphi:{\mathcal{D}}(\boldsymbol{\mu})\to{\mathbb{C}}$. Note that, if $f\in
F_{s}^{2}(\boldsymbol{\mu})$, then, due to Theorem 6.1, $Af$ has the
functional representation

$\left<Af,z_{\lambda}\right>=\left<\psi,A^{*}z_{\lambda}\right>=\varphi(\lambda)\psi(\lambda)\quad\text{
for all }\ \lambda\in{\mathcal{D}}(\boldsymbol{\mu}).$

In particular, $\left<A(1),z_{\lambda}\right>=\varphi(\lambda)$. Therefore
$\varphi$ can be identified with $A(1)\in F_{s}^{2}(\boldsymbol{\mu})$ and the
relation above shows that
$\varphi\psi\in{\mathbb{H}}^{2}({\mathcal{D}}(\boldsymbol{\mu}))$ for any
$\psi\in F_{s}^{2}(\boldsymbol{\mu})$. Therefore,
$A\in{\mathcal{M}}(F_{s}^{2}(\boldsymbol{\mu}))$ and
$\varphi\in{\mathcal{M}}({\mathbb{H}}^{2}({\mathcal{D}}(\boldsymbol{\mu})))$.
Since the Hardy algebra $F_{s}^{\infty}(\boldsymbol{\mu})$ is identified with
the multiplier algebra
${\mathcal{M}}({\mathbb{H}}^{2}({\mathcal{D}}(\boldsymbol{\mu})))$, the proof
is complete. ∎

##  7\. Functional calculus for commuting $n$-tuples of operators

Throughout this section, we assume that
$\boldsymbol{\mu}=\\{\mu_{\beta}\\}_{|\beta|\geq 1}$ is a weight sequence of
strictly positive numbers satisfying the boundedness conditions (4.1) and
(4.2) and having the property that ${\mathcal{D}}(\boldsymbol{\mu})$ is a
subset of ${\mathbb{C}}^{n}$ containing a neighborhood of the origin. Let
$B_{1},\ldots,B_{n}$ be the compressions of $W_{1},\ldots,W_{n}$ to the
symmetric weighted Fock space $F_{s}^{2}(\boldsymbol{\mu})$, respectively. We
recall that $F_{s}^{\infty}(\boldsymbol{\mu})$ is the $w^{*}$-closed non-self-
adjoint algebra generated by the operators $B_{1},\ldots,B_{n}$ and the
identity, which is identified with the multiplier algebra
${\mathcal{M}}({\mathbb{H}}^{2}({\mathcal{D}}(\boldsymbol{\mu})))$. Under this
identification, the $n$-tuple $(B_{1},\ldots,B_{n})$ is unitarily equivalent
to $(M_{\lambda_{1}},\ldots,M_{\lambda_{n}})$, where $M_{\lambda_{i}}$ is the
multiplication on ${\mathbb{H}}^{2}({\mathcal{D}}(\boldsymbol{\mu}))$ by the
coordinate function $\lambda_{i}$.

We recall from Section 5 that

$\Lambda_{\bf k}:=\\{\alpha\in{\mathbb{F}}_{n}^{+}:\
\lambda_{\alpha}=\lambda^{\bf k}\text{ for all
}\lambda\in{\mathbb{C}}^{n}\\},\qquad{\bf k}\in{\mathbb{N}}_{0}^{n},$

and

$y^{(\bf k)}:=\frac{1}{\omega_{\bf k}}\sum_{\alpha\in\Lambda_{\bf
k}}\frac{1}{\boldsymbol{\mu}(\alpha,g_{0})}e_{\alpha}\in
F^{2}(H_{n}),\quad\text{ where }\ \omega_{\bf k}:=\sum_{\alpha\in\Lambda_{\bf
k}}\frac{1}{\boldsymbol{\mu}(\alpha,g_{0})^{2}}.$

The set $\\{y^{(\bf k)}:\ {\bf k}\in{\mathbb{N}}_{0}^{n}\\}$ consists of
orthogonal vectors in $F^{2}(H_{n})$ and $\|y^{(\bf
k)}\|=\frac{1}{\sqrt{\omega_{\bf k}}}$, and the symmetric weighted Fock space
$F_{s}^{2}(\boldsymbol{\mu})$ is the closed span of these vectors.

We denote by ${\mathcal{D}}^{c}_{\boldsymbol{\mu}}({\mathcal{H}})$ the set all
all $n$-tuples
$X=(X_{1},\ldots,X_{n})\in{\mathcal{D}}_{\boldsymbol{\mu}}({\mathcal{H}})$
such that $X_{i}X_{j}=X_{j}X_{i}$ for any $i,j\in\\{1,\ldots,n\\}$. Note that
$X\in{\mathcal{D}}_{\boldsymbol{\mu}}({\mathcal{H}})$ if and only if
$\sum_{{\bf k}\in{\mathbb{N}}_{0}^{n}}\omega_{\bf k}T^{\bf k}{T^{\bf k}}^{*}$
is WOT-convergent. Let ${\mathcal{A}}_{s}(\boldsymbol{\mu})$ be the norm
closed non-self-adjoint algebra generated by $B_{1},\ldots,B_{n}$ and the
identity.

######  Theorem 7.1.

If $T=(T_{1},\ldots,T_{n})\in B({\mathcal{H}})^{n}$ is an $n$-tuple of
commuting operators and there is a positive invertible operator $Q\in
B({\mathcal{H}})$ and positive constants $0<a\leq b$ such that

$aI\leq\sum_{{\bf k}\in{\mathbb{N}}_{0}^{n}}\omega_{\bf k}T^{\bf k}Q{T^{\bf
k}}^{*}\leq bI,$

then $(T_{1},\ldots,T_{n})$ is jointly similar to

$(P_{\mathcal{M}}(B_{1}\otimes
I_{\mathcal{H}})|_{\mathcal{M}},\ldots,P_{\mathcal{M}}(B_{n}\otimes
I_{\mathcal{H}})|_{\mathcal{M}}),$

where ${\mathcal{M}}\subset F_{s}^{2}(\boldsymbol{\mu})\otimes{\mathcal{H}}$
is a joint invariant subspace under the operators $B_{i}^{*}\otimes
I_{\mathcal{H}}$, $i\in\\{1,\ldots,n\\}$. In this case, we have $r(T)\leq
r(B)$ and the map $\Phi_{T}:{\mathcal{A}}_{s}(\boldsymbol{\mu})\to
B(F^{2}(H_{n})$ defined by

$\Phi_{T}(p(B_{1},\ldots,B_{n})):=p(T_{1},\ldots,T_{n})$

is completely bounded and $\|\Phi_{T}\|_{cb}\leq\sqrt{\frac{b}{a}}$.

###### Proof.

A close look at the proof of Theorem 2.1 reveals that, if $T_{1},\ldots,T_{n}$
are commuting operators, then the operator
$K_{\boldsymbol{\mu}}:{\mathcal{H}}\to F^{2}(H_{n})\otimes{\mathcal{D}}$ has
the range in $F_{s}^{2}(\boldsymbol{\mu})\otimes{\mathcal{D}}$. Indeed, we
have

$K_{\boldsymbol{\mu}}h=\sum_{{\bf k}\in{\mathbb{N}}_{0}^{n}}\omega_{\bf
k}y^{(\bf k)}\otimes Q^{1/2}{T^{\bf k}}^{*}h\in
F_{s}^{2}(\boldsymbol{\mu})\otimes{\mathcal{D}}.$

Moreover, since $K_{\boldsymbol{\mu}}T_{i}^{*}=(W_{i}^{*}\otimes
I)K_{\boldsymbol{\mu}}$ and
$W_{i}^{*}|_{F^{2}_{s}(\boldsymbol{\mu})}=B_{i}^{*}$ for any
$i\in\\{1,\ldots,n\\}$, one can see that the subspace
${\mathcal{M}}:=K_{\boldsymbol{\mu}}{\mathcal{H}}\subset
F_{s}^{2}(\boldsymbol{\mu})\otimes{\mathcal{D}}$ is invariant under all
$B_{i}^{*}\otimes I$ and, consequently,
$K_{\boldsymbol{\mu}}T_{i}^{*}=(B_{i}^{*}\otimes I)K_{\boldsymbol{\mu}}$.
Defining $X:{\mathcal{H}}\to{\mathcal{M}}$ by $Xh:=K_{\boldsymbol{\mu}}h$,
$h\in{\mathcal{H}}$, it is clear that $X$ is an invertible operator and
$T_{i}^{*}=X^{-1}(B_{i}^{*}\otimes I)|_{\mathcal{M}}X$ for any
$i\in\\{1,\ldots,n\\}$. The remaining of the proof is similar to that of
Theorem 2.1. This completes the proof. ∎

We remark that if $Q=I$ in Theorem 7.1, we obtain a Rota type similarity
result for the elements
$T\in{\mathcal{D}}^{c}_{\boldsymbol{\mu}}({\mathcal{H}})$. In this case, we
have $\|\Phi_{T}\|_{cb}\leq\left\|\sum_{{\bf
k}\in{\mathbb{N}}_{0}^{n}}\omega_{\bf k}T^{\bf k}{T^{\bf
k}}^{*}\right\|^{1/2}$.

In what follows we present a $w^{*}$-continuous
$F_{s}^{\infty}(\boldsymbol{\mu})$-functional calculus for the elements in the
commutative Reinhardt set
${\mathcal{D}}^{c}_{\boldsymbol{\mu}}({\mathcal{H}})$, where ${\mathcal{H}}$
is a separable Hilbert space.

######  Theorem 7.2.

Let
$T=(T_{1},\ldots,T_{n})\in{\mathcal{D}}^{c}_{\boldsymbol{\mu}}({\mathcal{H}})$
and let $\Psi_{T}:F_{s}^{\infty}(\boldsymbol{\mu})\to B({\mathcal{H}})$ be
defined by

$\Psi_{T}(\varphi(B))=\varphi(T)=:\text{\rm SOT-}\lim_{N\to\infty}\sum_{{\bf
k}\in{\mathbb{N}}_{0}^{n},|{\bf k}|\leq N}\left(1-\frac{|{\bf
k}|}{N+1}\right)c_{\bf k}T^{\bf k}.$

where $\varphi(B)\in F_{s}^{\infty}(\boldsymbol{\mu})$ has the Fourier
representation $\sum_{{\bf k}\in{\mathbb{N}}_{0}^{n}}c_{\bf k}B^{\bf k}$. Then
$\Psi_{T}$ has the following properties.

  1. (i)

$\Psi_{T}\left(\sum_{|\alpha|\leq
m}c_{\alpha}B_{\alpha}\right)=\sum_{|\alpha|\leq m}c_{\alpha}T_{\alpha},\qquad
m\in{\mathbb{N}}.$

  2. (ii)

$\Psi_{T}$ is sequentially WOT-(resp. SOT-)continuous.

  3. (iii)

$\Psi_{T}$ is a completely bounded algebra homomorphism and

$\|\Psi_{T}\|_{cb}\leq\left\|\sum_{{\bf k}\in{\mathbb{N}}_{0}^{n}}\omega_{\bf
k}T^{\bf k}{T^{\bf k}}^{*}\right\|^{1/2}.$

  4. (iv)

$\Psi_{T}$ is $w^{*}$-continuous.

  5. (v)

$r(\varphi(T))\leq r(\varphi(B))$ for any $\varphi(B)\in
F_{s}^{\infty}(\boldsymbol{\mu})$, where $r(X)$ denotes the spectral radius of
$X$.

###### Proof.

Applying Theorem 7.1 in the particular case when $Q=I$, we find a subspace
${\mathcal{M}}\subset F_{s}^{2}(\boldsymbol{\mu})\otimes{\mathcal{H}}$ which
is jointly invariant subspace under all the operators $B_{i}^{*}\otimes
I_{\mathcal{H}}$ and an invertible operator $X:{\mathcal{H}}\to{\mathcal{M}}$
such that $T_{i}=X^{*}P_{\mathcal{M}}(B_{i}\otimes
I)|_{\mathcal{M}}(X^{*})^{-1}$, $i\in\\{1,\ldots,n\\}$, and
$\|X\|=\left\|\sum_{{\bf k}\in{\mathbb{N}}_{0}^{n}}\omega_{\bf k}T^{\bf
k}{T^{\bf k}}^{*}\right\|^{1/2}$. Define the map
$\Phi_{T}:F_{s}^{\infty}(\boldsymbol{\mu})\to B({\mathcal{H}})$ by setting

$\Phi_{T}(\varphi(B)):=X^{*}P_{\mathcal{M}}(\varphi(B)\otimes
I_{\mathcal{H}})|_{\mathcal{M}}(X^{*})^{-1},\qquad\varphi(B)\in
F_{s}^{\infty}(\boldsymbol{\mu}).$

Following the lines of the proof of Theorem 5.1, one can similarly show that
the map $\Phi_{T}$ has all the required properties and coincides with
$\Psi_{T}$. In addition, we need to use Theorem 6.3 and the remarks preceding
it. ∎

######  Theorem 7.3.

If $T=(T_{1},\ldots,T_{n})\in B({\mathcal{H}})^{n}$ is an $n$-tuple of
commuting operators, then there exists a weight sequence
$\boldsymbol{\mu}=\\{\mu_{\beta}\\}_{|\beta|\geq 1}$ with the following
properties:

  1. (i)

$\|\sum_{|\alpha|=k}B_{\alpha}B_{\alpha}^{*}\|^{1/2}\leq(k+1)\|\sum_{|\alpha|=k}T_{\alpha}T_{\alpha}^{*}\|^{1/2}$
for any $k\in{\mathbb{N}}$;

  2. (ii)

$r(B)=r(T)$;

  3. (iii)

$(T_{1},\ldots,T_{n})$ is jointly similar to

$(P_{\mathcal{M}}(B_{1}\otimes
I_{\mathcal{H}})|_{\mathcal{M}},\ldots,P_{\mathcal{M}}(B_{n}\otimes
I_{\mathcal{H}})|_{\mathcal{M}}),$

where ${\mathcal{M}}\subset F_{s}^{2}(\boldsymbol{\mu})\otimes{\mathcal{H}}$
is a joint invariant subspace under the operators $B_{i}^{*}\otimes
I_{\mathcal{H}}$, $i\in\\{1,\ldots,n\\}$;

  4. (iv)

$r(p(T_{1},\ldots,T_{n}))\leq r(p(B_{1},\ldots,B_{n}))$ and the map
$\Phi_{T}:{\mathcal{A}}_{s}(\boldsymbol{\mu})\to B(F^{2}(H_{n})$ defined by

$\Phi_{T}(p(B_{1},\ldots,B_{n})):=p(T_{1},\ldots,T_{n})$

is completely bounded and $\|\Phi_{T}\|_{cb}\leq\frac{\pi}{\sqrt{6}}$.

If $T=(T_{1},\ldots,T_{n})$ is a nilpotent $n$-tuple of order $m\geq 2$, then
there is a weighted commuting multi-shift $B=(B_{1},\ldots,B_{n})$ which is
nilpotent of order $m$ such that all the properties above hold and

$\|\Phi_{T}\|_{cb}\leq\sqrt{\sum_{k=0}^{m-1}\frac{1}{(k+1)^{2}}}.$

###### Proof.

Assume that $T=(T_{1},\ldots,T_{n})$ is not nilpotent. Consider the weight
sequence $\boldsymbol{\mu}=\\{\mu_{\beta}\\}_{|\beta|\geq 1}$ defined in the
proof of Theorem 2.5 and let $W=(W_{1},\ldots,W_{n})$ and
$B=(B_{1},\ldots,B_{n})$ be the corresponding multi-shifts associated with
$\boldsymbol{\mu}$. According to the same proof, we have $r(W)=r(T)$ and

$\left\|\sum_{{\bf k}\in{\mathbb{N}}_{0}^{n}}\omega_{\bf k}T^{\bf k}{T^{\bf
k}}^{*}\right\|\leq\sum_{k=0}^{\infty}\frac{1}{(k+1)^{2}}=\frac{\pi^{2}}{6}.$

Since $T_{1},\ldots,T_{n}$ are commuting, we apply Theorem 7.1 when $Q=I$ and
deduce (iii) and (iv). In particular, we have $r(T)\leq r(B)$. On the other
hand, since $B_{i}^{*}=W_{i}^{*}|_{F_{s}^{2}(\boldsymbol{\mu})}$, it is clear
that
$\|\sum_{|\alpha|=k}B_{\alpha}B_{\alpha}^{*}\|\leq\|\sum_{|\alpha|=k}W_{\alpha}W_{\alpha}^{*}\|$
for any $k\in{\mathbb{N}}$ and, consequently, $r(B)\leq r(W)$. Using again
Theorem 2.5, one can complete the proof. The case when $T$ is a commuting
nilpotent tuple can be treated in a similar manner. ∎

######  Theorem 7.4.

If $T=(T_{1},\ldots,T_{n})\in B({\mathcal{H}})^{n}$ is an $n$-tuple of
commuting operators, then there exists a weight sequence
$\boldsymbol{\mu}=\\{\mu_{\beta}\\}_{|\beta|\geq 1}$ such that

  1. (i)

$r(B)=r(T)$;

  2. (ii)

the conditions (4.1), (4.2) are satisfied;

  3. (iii)

if $T$ is not nilpotent, ${\mathcal{D}}(\boldsymbol{\mu})$ contains a
neighborhood of the origin;

  4. (iv)

the map $\Psi_{T}:F_{s}^{\infty}(\boldsymbol{\mu})\to B({\mathcal{H}})$
defined by

$\Psi_{T}(\varphi(B))=\varphi(T)=:\text{\rm SOT-}\lim_{N\to\infty}\sum_{{\bf
k}\in{\mathbb{N}}_{0}^{n},|{\bf k}|\leq N}\left(1-\frac{|{\bf
k}|}{N+1}\right)c_{\bf k}T^{\bf k},$

where $\varphi(B)\in F_{s}^{\infty}(\boldsymbol{\mu})$ has the Fourier
representation $\sum_{{\bf k}\in{\mathbb{N}}_{0}^{n}}c_{\bf k}B^{\bf k}$, has
all the properties listed in Theorem 7.2 and
$\|\Psi_{T}\|_{cb}\leq\frac{\pi}{\sqrt{6}}$.

When $T=(T_{1},\ldots,T_{n})$ is a nilpotent $n$-tuple of order $m\geq 2$, we
have

$\|\Psi_{T}\|_{cb}\leq\sqrt{\sum_{k=0}^{m-1}\frac{1}{(k+1)^{2}}}.$

###### Proof.

Assume that $r(T)>0$. According to Theorem 7.3, we find a weight sequence
$\boldsymbol{\mu}=\\{\mu_{\beta}\\}_{|\beta|\geq 1}$ such that items (i) and
(ii) hold. Due to Corollary (5.3), we have
$({\mathbb{C}}^{n})_{r(T)}\subset\sigma_{p}(W_{1}^{*},\ldots,W_{n}^{*})={\mathcal{D}}(\boldsymbol{\mu})$
which proves item (iii). Now, we can apply Theorem 7.2 and Theorem 7.3 to
deduce item (iv). The case when $T$ is nilpotent can be treated in a similar
manner. ∎

Recall that ${\rm Hol}_{0}(Z)$ denotes the algebra of all free holomorphic
functions $f$ satisfying condition (5.4).

######  Theorem 7.5.

Let $T=(T_{1},\ldots,T_{n})\in B({\mathcal{H}})^{n}$ be a quasi-nilpotent
commuting $n$-tuple of operators. Then there exists a quasi-nilpotent
commuting weighted multi-shift $B=(B_{1},\ldots,B_{n})$ with the following
properties.

  1. (i)

$\|f(T_{1},\ldots,T_{n})\|\leq\frac{\pi}{\sqrt{6}}\|f(B_{1},\ldots,B_{n})\|$
for any $f\in{\rm Hol}_{0}(Z)$.

  2. (ii)

$r(f(T_{1},\ldots,T_{n}))\leq r(f(B_{1},\ldots,B_{n}))$ for any $f\in{\rm
Hol}_{0}(Z)$.

  3. (iii)

Let $\\{f_{k}\\},f\in{\rm Hol}_{0}(Z)$ be such that
$\\{\|f_{k}(B_{1},\ldots,B_{n})\|\\}_{k}$ is a bounded sequence and

$f_{k}(B_{1},\ldots,B_{n})\to f(B_{1},\ldots,B_{n}),\qquad\text{as }\
k\to\infty,$

in the operator norm (resp. WOT-, $w^{*}$, SOT-) topology, then
$f_{k}(T_{1},\ldots,T_{n})\to f(T_{1},\ldots,T_{n})$ in the corresponding
topology, respectively.

###### Proof.

The proof is similar to the proof of Theorem 5.7, but uses Theorem 7.3. ∎

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