Invertibility of functions of operators and existence of hyperinvariant subspaces

TL;DR

This paper investigates conditions under which certain operators possess nontrivial hyperinvariant subspaces, focusing on invertibility of functions of operators involving singular inner functions.

Contribution

It establishes new criteria linking invertibility of functions of operators to the existence of hyperinvariant subspaces for polynomially bounded operators.

Findings

Invertibility of $ heta(T)$ implies hyperinvariant subspaces under certain conditions

Provides conditions involving singular inner functions and polynomially bounded operators

Advances understanding of the structure of operators with hyperinvariant subspaces

Abstract

Let $T$ be an absolutely continuous polynomially bounded operator, and let $\theta$ be a singular inner function. It is shown that if $\theta(T)$ is invertible and some additional conditions are fulfilled, then $T$ has nontrivial hyperinvariant subspaces.

Equations

$$\Vert p(T)\Vert \le M\sup \{|p(z)|:|z|\le 1\}$$

$${\theta }_{\mu }(z)=\exp {\int }_{\mathbb{T}}\frac{z+\zeta }{z-\zeta }\text{d}\mu (\zeta )(z\in \mathbb{D}).$$

$$D(\lambda )=\{z\in \mathbb{C}:|z-\lambda |<1-|\lambda |\}.$$

$$\frac{1-|z{|}^2}{|\zeta -z{|}^2}$$

$$\text{ and }\frac{1-|z{|}^2}{|\zeta -z{|}^2}$$

$$|{\theta }_{\mu }(z)|\ge \exp \frac{r\mu (E)}{r-1}\text{ for every }z\in \mathbb{D}\setminus \bigcup _{n\in \mathfrak{N},k=1,2}D(r{\zeta }_{kn}).$$

$$|b_{\lambda }(z)|\ge \frac{1}{3}\text{ for every }z\in \mathbb{D}\setminus D(\lambda ).$$

$$\begin{array}{c}\Gamma \cap \mathbb{T}=\{{\zeta }_{\Gamma 1},{\zeta }_{\Gamma 2}\},{\zeta }_{\Gamma 1}\ne {\zeta }_{\Gamma 2},\text{ and there exists }\epsilon >0\text{ such that }\\ \Gamma \cap \{z\in \mathbb{C}:|z-{\zeta }_{\Gamma l}|\le \epsilon ,|z|\le 1\}\text{ and }\Gamma \cap \{z\in \mathbb{C}:|z-{\zeta }_{\Gamma l}|\le \epsilon ,|z|\ge 1\}\\ \text{ are segments nontangential to }\mathbb{T}\text{ at }{\zeta }_{\Gamma l}(l=1,2).\end{array}$$

$$\Gamma \setminus \{{\zeta }_{\Gamma 1},{\zeta }_{\Gamma 2}\}\subset \mathbb{C}\setminus ({\Gamma }^{\prime }\cup {\Omega }_{{\Gamma }^{\prime }}),{\Gamma }^{\prime }\setminus \{{\zeta }_{{\Gamma }^{\prime }1},{\zeta }_{{\Gamma }^{\prime }2}\}\subset \mathbb{C}\setminus (\Gamma \cup {\Omega }_{\Gamma }).$$

$$\underset{\lambda \in \Gamma \setminus \{{\zeta }_{\Gamma 1},{\zeta }_{\Gamma 2}\}}{\sup }|p(\lambda )|\Vert (T-\lambda I{)}^{-1}\Vert <\infty ,\underset{\lambda \in {\Gamma }^{\prime }\setminus \{{\zeta }_{{\Gamma }^{\prime }1},{\zeta }_{{\Gamma }^{\prime }2}\}}{\sup }|q(\lambda )|\Vert (T-\lambda I{)}^{-1}\Vert <\infty .$$

$$A=\frac{1}{2\pi i}{\int }_{\Gamma }p(\lambda )(T-\lambda I{)}^{-1}\text{d}\lambda ,A^{\prime }=\frac{1}{2\pi i}{\int }_{{\Gamma }^{\prime }}q(\lambda )(T-\lambda I{)}^{-1}\text{d}\lambda .$$

$$\varphi (A{|}_{\mathcal{E}})=\frac{1}{2\pi i}{\int }_{\Gamma }p(\lambda )(\xi -\lambda {)}^{-1}\text{d}\lambda \ne 0.$$

$$\Lambda=\sigma(T)\cup\bigl{\{}\lambda\in\mathbb{D}:\ \ \|(T-\lambda I)^{-1}\|\geq C/(1-|\lambda|)^{k}\bigr{\}}$$

$$\mathcal{Z}=\bigl{\{}\zeta\in\sigma(T)\cap\mathbb{T}:\ \ \sup\{r\in[0,1):r\zeta\in\Lambda\}<1\bigr{\}}.$$

$$F(\zeta ,a,b)=\{r\zeta :a\le r<1\}\cup \{z\in \mathbb{D}:|z-a\zeta |<b\}.$$

$${\cup }_{j=1}^{10}F({\xi }_j,a,b)\subset \mathbb{D}\setminus \Lambda \text{ and }{\cap }_{j=1}^{10}F({\xi }_j,a,b)\ne \varnothing .$$

$$({\Gamma }_1\cup {\Gamma }_1^{\prime }\cup {\Gamma }_2\cup {\Gamma }_2^{\prime })\cap \mathbb{D}\subset {\cup }_{j=1}^{10}F({\xi }_j,a,b),$$

$$\displaystyle\zeta_{\Gamma_{1}1}=\xi_{4},\ \ \ \zeta_{\Gamma_{1}2}=\xi_{10},\ \ \ \ \ \$$

$$\displaystyle\zeta_{\Gamma_{2}1}=\xi_{5},\ \ \ \zeta_{\Gamma_{2}2}=\xi_{9},\ \ \ \ \ \$$

$${\xi }_5,{\xi }_6,{\xi }_7,{\xi }_8,{\xi }_9\in {\Omega }_{{\Gamma }_1},{\xi }_2\in {\Omega }_{{\Gamma }_1^{\prime }},{\xi }_1,{\xi }_2,{\xi }_3,{\xi }_4,{\xi }_{10}\in {\Omega }_{{\Gamma }_2},{\xi }_7\in {\Omega }_{{\Gamma }_2^{\prime }},$$

$$(\mathrm{clos}\mathbb{D}\setminus {\Omega }_{{\Gamma }_1})\cap (\mathrm{clos}\mathbb{D}\setminus {\Omega }_{{\Gamma }_2})=\varnothing ,$$

$$\Vert (T-\lambda {)}^{-1}(I-\overline{\lambda }T)\Vert <C\text{ for all }\lambda \in \mathbb{D}\text{ such that }|\phi (\lambda )|\le c.$$

$$\Vert (T-\lambda I{)}^{-1}x\Vert \le C\Vert (I-\overline{\lambda }T{)}^{-1}x\Vert \text{ and }\Vert (T-\lambda I{)}^{-1}\Vert <CM/(1-|\lambda |)$$

$$(\phi (T)-a{)}^{-1}=\phi (T{)}^{-1}(I-a\phi (T{)}^{-1}{)}^{-1},$$

$$\Vert (\phi (T)-a{)}^{-1}\Vert <\frac{1}{c}\cdot \frac{1}{1-c\Vert \phi (T{)}^{-1}\Vert }.$$

$$\Vert \psi {\Vert }_{\infty }=\Vert {\beta }_{\lambda }\psi {\Vert }_{\infty }=\Vert \phi -\phi (\lambda ){\Vert }_{\infty }\le \Vert \phi {\Vert }_{\infty }+c.$$

$${\beta }_{\lambda }(T{)}^{-1}=(\phi (T)-\phi (\lambda ){)}^{-1}\psi (T),$$

$$\Vert {\beta }_{\lambda }(T{)}^{-1}\Vert <M\frac{\Vert \phi {\Vert }_{\infty }+c}{c(1-c\Vert \phi (T{)}^{-1}\Vert )}=:C.$$

$$(T-\lambda I{)}^{-1}={\beta }_{\lambda }(T{)}^{-1}(I-\overline{\lambda }T{)}^{-1}.$$

$$\sigma (\phi (T))=\phi (\sigma (T)),$$

$$\mathcal{Z}_{0}=\bigl{\{}\zeta\in\sigma(T)\cap\mathbb{T}:\ \ \theta_{\mu}(z)\to 0\ \text{ when }\ z\to\zeta\ \text{ nontangentially}\bigr{\}}$$

Full text

1. Introduction

Let $\mathcal{H}$ be a (complex, separable, infinite dimensional) Hilbert space, and let $\mathcal{L}(\mathcal{H})$ be the algebra of all (linear, bounded) operators acting on $\mathcal{H}$. A (closed) subspace $\mathcal{M}$ of $\mathcal{H}$ is called invariant for an operator $T\in\mathcal{L}(\mathcal{H})$, if $T\mathcal{M}\subset\mathcal{M}$, and $\mathcal{M}$ is called hyperinvariant for $T$ if $A\mathcal{M}\subset\mathcal{M}$ for all $A\in\mathcal{L}(\mathcal{H})$ such that $AT=TA$. The complete lattice of all invariant (resp., hyperinvariant) subspaces of $T$ is denoted by $\operatorname{Lat}T$ (resp., by $\operatorname{Hlat}T$). The hyperinvariant subspace problem is the question whether $\operatorname{Hlat}T$ is nontrivial (i.e., $\operatorname{Hlat}T\neq\{\{0\},\mathcal{H}\}$) whenever $T$ is not a scalar multiple of the identity operator $I$.

The operator $T\in\mathcal{L}(\mathcal{H})$ is called polynomially bounded, if there exists a constant $M$ such that

[TABLE]

for every (analytic) polynomial $p$. For every polynomially bounded operator $T\in\mathcal{L}(\mathcal{H})$ there exist $\mathcal{H}_{a}$, $\mathcal{H}_{s}\in\operatorname{Hlat}T$ such that $\mathcal{H}=\mathcal{H}_{a}\dotplus\mathcal{H}_{s}$, $T|_{\mathcal{H}_{s}}$ is similar to a singular unitary operator, and $T|_{\mathcal{H}_{a}}$ has an $H^{\infty}$-functional calculus, that is, $T|_{\mathcal{H}_{a}}$ is an absolutely continuous (a.c.) polynomially bounded operator [Ml], [K16]. Thus, if $\mathcal{H}_{s}\neq\{0\}$ and $T\neq cI$, then $T$ has nontrivial hyperinvariant subspaces. For the existence of invariant subspaces of polynomially bounded operators see [Ré].

The operator $T\in\mathcal{L}(\mathcal{H})$ is called a contraction, if $\|T\|\leq 1$. Every contraction is polynomially bounded with constant $1$ by the von Neumann inequality (see, for example, [SFBK, Proposition I.8.3]). (It is well known that the converse is not true, see [Pi] for the first example of a polynomially bounded operator which is not similar to a contraction.)

It is well known that if the spectrum $\sigma(T)$ of an operator $T$ is not connected, then nontrivial hyperinvariant subspaces of $T$ can be found by using the Riesz–Dunford functional calculus, see, for example [RR, Theorem 2.10]. A similar method can be applied, if an operator $T$ has sufficiently rich spectrum and appropriate estimate for the norm of the resolvent [A], [CP, Sec. 4.1]. Using such a method, we show that an a.c. polynomially bounded operator $T$ has nontrivial hyperinvariant subspaces, if $\theta(T)$ is invertible for a singular inner function $\theta$ and some additional conditions are fulfilled (Sec. 2). Moreover, in this case $T$ cannot be quasianalytic. Examples of quasianalytic operators $T$ for which $\theta(T)$ is invertible for some inner functions $\theta$ are given in Sec. 3. (See the beginning of Sec. 3 for references about quasianalyticity.) In Sec. 4 it is shown that if $\varphi(T)$ is invertible for some weighted shift $T$ and function $\varphi$, then $T$ is similar to the simple bilateral shift.

Symbols $\mathbb{D}$, $\operatorname{clos}\mathbb{D}$, and $\mathbb{T}$ denote the open unit disc, the closed unit disc, and the unit circle, respectively. The normalized Lebesgue measure on $\mathbb{T}$ is denoted by $m$.

For any finite, positive, singular (with respect to $m$) Borel measure $\mu$ on $\mathbb{T}$ define a function $\theta_{\mu}$ by the formula

[TABLE]

The function $\theta_{\mu}$ is a singular inner function. Recall that $\theta_{\mu}$ has nontangential boundary values equal to zero a.e. with respect to $\mu$ (see, for example, [Gar, Theorem II.6.2]). As usual, the Dirac measure at a point $\zeta$ is denoted by $\delta_{\zeta}$.

For $\lambda\in\mathbb{D}$ set

[TABLE]

The following simple lemmas are given for convenience of references.

Lemma 1.1**.**

Let $0<r<1$, and let $\zeta\in\mathbb{T}$. Then

[TABLE]

Lemma 1.2**.**

Suppose that $E\subset\mathbb{T}$ is a non-empty closed set, $m(E)=0$, $\mathbb{T}\setminus E=\cup_{n\in\mathfrak{N}}\mathcal{J}_{\zeta_{1n},\zeta_{2n}}$, where $\mathfrak{N}\subset\mathbb{N}$, $\mathcal{J}_{\zeta_{1n},\zeta_{2n}}$ is the open subarc of $\mathbb{T}$ with endpoints $\zeta_{1n}$,$\zeta_{2n}$, $\mathcal{J}_{\zeta_{1n},\zeta_{2n}}\cap\mathcal{J}_{\zeta_{1l},\zeta_{2l}}=\emptyset$ for $n\neq l$. Furthermore, let $0<r<1$, and let $\mu$ be a finite positive Borel measure on $E$. Then

[TABLE]

Lemma 1.3**.**

Let $\lambda\in\mathbb{D}\setminus\{0\}$. Set $b_{\lambda}(z)=\frac{|\lambda|}{\lambda}\frac{\lambda-z}{1-\overline{\lambda}z}\ \ (z\in\mathbb{D}).$ Then

[TABLE]

2. Estimates of the resolvent

The main result of this section (Theorem 2.5) is the following. Let $T$ be an a.c. polynomially bounded operator, and let $\theta$ be a singular inner function. If $\theta(T)$ is invertible, then, under some additional conditions, $T$ has nontrivial hyperinvariant subspaces.

Suppose that $\Gamma$ is a simple closed curve,

[TABLE]

Let $\Omega_{\Gamma}$ be the bounded component of $\mathbb{C}\setminus\Gamma$.

The following lemmas are well known, see, for example, [A], [B, Lemma 3.1], [T, Lemma 6], [CP, Sec. 4.1]. We give proofs to emphasize some details.

Lemma 2.1**.**

Suppose that $\Gamma$ and $\Gamma^{\prime}$ are two simple closed rectifiable curves, $\Gamma$ and $\Gamma^{\prime}$ satisfy (2.1), and

[TABLE]

Suppose that $T\in\mathcal{L}(\mathcal{H})$ has the following properties:

1. (i)

there exist $C>0$ and $k\in\mathbb{N}$ such that \|(T-\lambda I)^{-1}\|\leq C/\bigl{|}1-|\lambda|\bigr{|}^{k} for all $\lambda\in\Gamma\setminus\{\zeta_{\Gamma 1},\zeta_{\Gamma 2}\}$ and all $\lambda\in\Gamma^{\prime}\setminus\{\zeta_{\Gamma^{\prime}1},\zeta_{\Gamma^{\prime}2}\}$; 2. (ii)

$\sigma(T)\cap\Omega_{\Gamma}\neq\emptyset$* and $\sigma(T)\cap\Omega_{\Gamma^{\prime}}\neq\emptyset$.*

Then there exist $\mathcal{M}$, $\mathcal{M}^{\prime}\in\operatorname{Hlat}T$ such that $\mathcal{M}\neq\{0\}$, $\mathcal{M}^{\prime}\neq\{0\}$, $\sigma(T|_{\mathcal{M}})\subset\mathbb{C}\setminus\Omega_{\Gamma}$, and $\sigma(T|_{\mathcal{M}^{\prime}})\subset\mathbb{C}\setminus\Omega_{\Gamma^{\prime}}$.

Proof.

Set $p(\lambda)=(\lambda-\zeta_{\Gamma 1})^{k}(\lambda-\zeta_{\Gamma 2})^{k}$, $q(\lambda)=(\lambda-\zeta_{\Gamma^{\prime}1})^{k}(\lambda-\zeta_{\Gamma^{\prime}2})^{k}$, $\lambda\in\mathbb{C}$. Since the segments defined in (2.1) are nontangential to $\mathbb{T}$ at $\zeta_{\Gamma l}$, $\zeta_{\Gamma^{\prime}l}$ $\ (l=1,2)$, we have that

[TABLE]

Put

[TABLE]

Let $\{0\}\neq\mathcal{E}\in\operatorname{Hlat}T$. Let ${\mathcal{A}}_{\mathcal{E}}\subset\mathcal{L}(\mathcal{E})$ be a maximal commutative Banach algebra such that $T|_{\mathcal{E}}$, $(T|_{\mathcal{E}}-\lambda I_{\mathcal{E}})^{-1}\in{\mathcal{A}}_{\mathcal{E}}$ for all $\lambda\in\mathbb{C}\setminus\sigma(T|_{\mathcal{E}})$. We infer from (2.3) that $A|_{\mathcal{E}}$, $A^{\prime}|_{\mathcal{E}}\in{\mathcal{A}}_{\mathcal{E}}$. Let $\xi\in\Omega_{\Gamma}$. If $\xi\in\sigma(T|_{\mathcal{E}})$, then there exists a continuous algebra homomorphism $\phi\colon{\mathcal{A}}_{\mathcal{E}}\to\mathbb{C}$ such that $\phi(T|_{\mathcal{E}})=\xi$. Then

[TABLE]

Applying (2.4) with $\mathcal{E}=\mathcal{H}$, we conclude that $A\neq\mathbb{O}$. Similarly, $A^{\prime}\neq\mathbb{O}$.

Put $\mathcal{M}=\ker A$ and $\mathcal{M}^{\prime}=\ker A^{\prime}$. Since $A$ and $A^{\prime}$ commute with all $B\in\mathcal{L}(\mathcal{H})$ such that $BT=TB$, we have $\mathcal{M}$, $\mathcal{M}^{\prime}\in\operatorname{Hlat}T$. We prove that $AA^{\prime}=A^{\prime}A=\mathbb{O}$ exactly as in [B, Lemma 3.1] or [CP, Sec. 4.1], and we obtain from the latter equality that $\mathcal{M}\neq\{0\}$ and $\mathcal{M}^{\prime}\neq\{0\}$.

Let $\xi\in\Omega_{\Gamma}$. If $\xi\in\sigma(T|_{\mathcal{M}})$, then, by (2.4) applied with $\mathcal{E}=\mathcal{M}$, there exists an algebra homomorphism $\phi\colon{\mathcal{A}}_{\mathcal{M}}\to\mathbb{C}$ such that $\phi(A|_{\mathcal{M}})\neq 0$, a contradiction with the definition of $\mathcal{M}$. Therefore, $\sigma(T|_{\mathcal{M}})\cap\Omega_{\Gamma}=\emptyset.$ Similarly, $\sigma(T|_{\mathcal{M}^{\prime}})\cap\Omega_{\Gamma^{\prime}}=\emptyset.$ ∎

Lemma 2.2**.**

Suppose that $C>0$, $k\in\mathbb{N}$, and an operator $T$ is given such that $\sigma(T)\subset\operatorname{clos}\mathbb{D}$ and $\|(T-\lambda I)^{-1}\|\leq C/(|\lambda|-1)^{k}$ for all $\lambda\notin\operatorname{clos}\mathbb{D}$. Put

[TABLE]

and

[TABLE]

If $\mathcal{Z}$ is uncountable, then there exist $\mathcal{M}_{1}$, $\mathcal{M}_{2}\in\operatorname{Hlat}T$ such that $\mathcal{M}_{1}\neq\{0\}$, $\mathcal{M}_{2}\neq\{0\}$, and $\sigma(T|_{\mathcal{M}_{1}})\cap\sigma(T|_{\mathcal{M}_{2}})=\emptyset$.

Proof.

For $\zeta\in\mathbb{T}$ and $a$, $b\in(0,1)$ put

[TABLE]

Since $\mathcal{Z}$ is uncountable, there exist $a$, $b\in(0,1)$ and $\xi_{j}\in\mathcal{Z}$, $j=1,\ldots,10$, such that $\xi_{j}$ are arranged on $\mathbb{T}$ in counter clockwise order,

[TABLE]

We construct four simple closed rectifiable curves $\Gamma_{1}$, $\Gamma_{1}^{\prime}$, $\Gamma_{2}$, $\Gamma_{2}^{\prime}$ such that they satisfy (2.1) (they cross $\mathbb{T}$ along radial segments),

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and each of the pairs $\Gamma_{1}$, $\Gamma_{1}^{\prime}$ and $\Gamma_{2}$, $\Gamma_{2}^{\prime}$ satisfies (2.2).

Applying Lemma 2.1 to $\Gamma_{1}$, $\Gamma_{1}^{\prime}$, we obtain $\mathcal{M}_{1}\in\operatorname{Hlat}T$ such that $\mathcal{M}_{1}\neq\{0\}$, and $\sigma(T|_{\mathcal{M}_{1}})\subset\mathbb{C}\setminus\Omega_{\Gamma_{1}}$. Applying Lemma 2.1 to $\Gamma_{2}$, $\Gamma_{2}^{\prime}$, we obtain $\mathcal{M}_{2}\in\operatorname{Hlat}T$ such that $\mathcal{M}_{2}\neq\{0\}$, and $\sigma(T|_{\mathcal{M}_{2}})\subset\mathbb{C}\setminus\Omega_{\Gamma_{2}}$. By [RR, Theorem 0.8], the spectrum of the restriction of an operator on its invariant subspace is contained in the polynomially convex hull of its spectrum. Therefore, $\sigma(T|_{\mathcal{M}_{l}})\subset\operatorname{clos}\mathbb{D}$ $\ (l=1,2)$. Consequently, $\sigma(T|_{\mathcal{M}_{l}})\subset\operatorname{clos}\mathbb{D}\setminus\Omega_{\Gamma_{l}}$ $\ (l=1,2)$. ∎

The investigation of a.c. polynomially bounded operators $T$ such that $\theta(T)$ is invertible for some inner function $\theta\in H^{\infty}$ is started in [K11, Proposition 4.4 and Theorem 5.4]. In particular, it is proved in [K11, Proposition 4.4] that $\mathbb{D}\not\subset\sigma(T)$ for such $T$. In [K11, Theorem 5.4] it is shown that if $\theta(T)$ is invertible for an inner function $\theta$, then $\theta(T)^{-1}$ belongs to the closure in the weak operator topology of the algebra of rational functions of $T$. Note that the assumption of quasianalyticity of $T$ from [K11, Proposition 4.4 and Theorem 5.4] actually is not used in their proofs, and the proofs work for a.c. polynomially bounded operators instead of a.c. contractions with minimal changes.

Lemma 2.3**.**

Suppose that $T\in\mathcal{L}(\mathcal{H})$ is an a.c. polynomially bounded operator, $\varphi\in H^{\infty}$, and $\varphi(T)$ is invertible. Then for every $0<c<1/\|\varphi(T)^{-1}\|$ there exists $C>0$ (which also depends on $\varphi$ and $T$) such that

[TABLE]

Consequently,

[TABLE]

for all $x\in\mathcal{H}$ and all $\lambda\in\mathbb{D}$ such that $|\varphi(\lambda)|\leq c$ (where $M$ is the polynomial bound of $T$).

Proof.

Let $a\in\mathbb{C}$, and let $|a|\leq c$. Then

[TABLE]

therefore,

[TABLE]

Suppose that $\lambda\in\mathbb{D}$, and $|\varphi(\lambda)|\leq c$. Then $\varphi-\varphi(\lambda)=\beta_{\lambda}\psi$, where $\beta_{\lambda}(z)=\frac{z-\lambda}{1-\overline{\lambda}z}$ $\ (z\in\mathbb{D})$, $\psi\in H^{\infty}$, and

[TABLE]

Since

[TABLE]

we obtain from (2.5), (2.6), and (2.7) that

[TABLE]

The remaining part of the conclusion of the lemma follows from the equality

[TABLE]

∎

Remark 2.4**.**

Lemma 2.3 can be considered as a certain type of spectral mapping theorem, cf. [BFP, Proof of Lemma 3.1], [Ma, 2.1(ii)]. It is well known that

[TABLE]

if $\varphi$ is a function analytic in a neighbourhood of $\sigma(T)$ and $\varphi(T)$ is defined by the Riesz–Dunford functional calculus for any (linear, bounded) operator $T$ (see, for example, [RR, Proposition 2.5], [CP, Theorem 1.2.25]). Furthermore, (2.8) holds if $T$ is a polynomially bounded operator and $\varphi$ is a function from $H^{\infty}$ which has continuous extension on $\sigma(T)$ [FM, Corollary 3.2]; see [JKP] for another proof. (For detailed definition of $\varphi(T)$, see [JKP]; decomposition on a.c. and singular parts of polynomially bounded operator is used.) In particular, (2.8) holds if $T$ is a polynomially bounded operator and $\varphi$ is a function from the disc algebra (that is, $\varphi$ is analytic in $\mathbb{D}$ and has continuous extension on $\mathbb{T}$), for proof see also [SFBK, Proposition IX.2.4]. But for $H^{\infty}$-functional calculus for a.c. polynomially bounded operators (and even for a.c. contractions) it is impossible to describe $\sigma(\varphi(T))$ in terms of $\varphi$ and $\sigma(T)$ only, see, for example, [BFP], [Ma], [Ru] and references therein.

Theorem 2.5**.**

Suppose that $T$ is an a.c. polynomially bounded operator and $\mu$ is a finite positive singular Borel measure on $\mathbb{T}$ such that $\mu(\sigma(T)\cap\mathbb{T})>0$ and $\mu(\{\zeta\})=0$ for every $\zeta\in\sigma(T)\cap\mathbb{T}$. Suppose that $\theta_{\mu}(T)$ is invertible. Then there exist $\mathcal{M}_{1}$, $\mathcal{M}_{2}\in\operatorname{Hlat}T$ such that $\mathcal{M}_{1}\neq\{0\}$, $\mathcal{M}_{2}\neq\{0\}$, and $\sigma(T|_{\mathcal{M}_{1}})\cap\sigma(T|_{\mathcal{M}_{2}})=\emptyset$.

Proof.

By [Gar, Theorem II.6.2], $\theta_{\mu}(z)\to 0$ when $z\to\zeta$ nontangentially for a.e. $\zeta\in\mathbb{T}$ with respect to $\mu$. Since $\mu(\sigma(T)\cap\mathbb{T})>0$ and $\mu(\{\zeta\})=0$ for every $\zeta\in\sigma(T)\cap\mathbb{T}$, the set

[TABLE]

is uncountable. Let $0<c<1/\|\theta_{\mu}(T)^{-1}\|$, and let $C$ be defined by $c$ in Lemma 2.3. Set

[TABLE]

and

[TABLE]

By Lemma 2.3, we have $\mathcal{Z}_{0}\subset\mathcal{Z}$ and so $\mathcal{Z}$ is uncountable. The conclusion of the theorem follows from Lemma 2.2. ∎

For $\zeta_{1}$, $\zeta_{2}\in\mathbb{T}$, $\zeta_{1}\neq\zeta_{2}$, denote by $J_{\zeta_{1},\zeta_{2}}$ the closed subarc of $\mathbb{T}$ with endpoints $\zeta_{1}$, $\zeta_{2}$ which connects $\zeta_{1}$ with $\zeta_{2}$ in counter clockwise order.

Theorem 2.6**.**

Suppose that $T$ is an a.c. polynomially bounded operator, $\sigma(T)=\mathbb{T}$, $\zeta_{1}$, $\zeta_{2}\in\mathbb{T}$, $\zeta_{1}\neq\zeta_{2}$, $\mu$ is a finite positive singular Borel measure on $\mathbb{T}$ such that $\mu(\{\zeta_{l}\})>0$ for $l=1,2$, and $\theta_{\mu}(T)$ is invertible. Then there exist $\mathcal{M}$, $\mathcal{M}^{\prime}\in\operatorname{Hlat}T$ such that $\mathcal{M}\neq\{0\}$, $\mathcal{M}^{\prime}\neq\{0\}$,

[TABLE]

Proof.

Let $0<c<1/\|\theta_{\mu}(T)^{-1}\|$. By Lemma 1.1, there exists $0<r<1$ such that

[TABLE]

Taking into account Lemma 2.3 and the assumption that $\sigma(T)\subset\mathbb{T}$, we conclude that there exist two simple closed rectifiable curves $\Gamma$ and $\Gamma^{\prime}$ such that $\Gamma$ and $\Gamma^{\prime}$ satisfy (2.1) with

[TABLE]

and

[TABLE]

for some $C>0$. It is easy to see that $\Gamma$, $\Gamma^{\prime}$ and $T$ satisfy the assumptions of Lemma 2.1. The conclusion of the theorem follows from the conclusion of Lemma 2.1.∎

Theorem 2.7**.**

Suppose that $T$ is an a.c. polynomially bounded operator, $\zeta_{1}$, $\zeta_{2}\in\mathbb{T}$, $\zeta_{1}\neq\zeta_{2}$, $\sigma(T)=J_{\zeta_{1},\zeta_{2}}$, $\zeta_{0}\in J_{\zeta_{1},\zeta_{2}}\setminus\{\zeta_{1},\zeta_{2}\}$, $\mu$ is a finite positive singular Borel measure on $\mathbb{T}$ such that $\mu(\{\zeta_{0}\})>0$, and $\theta_{\mu}(T)$ is invertible. Then there exist $\mathcal{M}$, $\mathcal{M}^{\prime}\in\operatorname{Hlat}T$ such that $\mathcal{M}\neq\{0\}$, $\mathcal{M}^{\prime}\neq\{0\}$,

[TABLE]

Proof.

Let $0<c<1/\|\theta_{\mu}(T)^{-1}\|$. By Lemma 1.1, there exists $0<r<1$ such that

[TABLE]

Taking into account Lemma 2.3 and the assumption that $\sigma(T)=J_{\zeta_{1},\zeta_{2}}$, we conclude that there exist two simple closed rectifiable curves $\Gamma$ and $\Gamma^{\prime}$ such that $\Gamma$ and $\Gamma^{\prime}$ satisfy (2.1) with

[TABLE]

and

[TABLE]

for some $C>0$. It is easy to see that $\Gamma$, $\Gamma^{\prime}$ and $T$ satisfy the assumptions of Lemma 2.1. Let $\mathcal{M}$ and $\mathcal{M}^{\prime}$ be the subspaces from the conclusion of Lemma 2.1. We have that $\sigma(T|_{\mathcal{M}})\subset\mathbb{C}\setminus\Omega_{\Gamma}$, and $\sigma(T|_{\mathcal{M}^{\prime}})\subset\mathbb{C}\setminus\Omega_{\Gamma^{\prime}}$. By [RR, Theorem 0.8], the spectrum of the restriction of an operator on its invariant subspace is contained in the polynomially convex hull of its spectrum. Therefore, $\sigma(T|_{\mathcal{M}})$, $\sigma(T|_{\mathcal{M}^{\prime}})\subset J_{\zeta_{1},\zeta_{2}}$. Consequently, $\sigma(T|_{\mathcal{M}})\subset(\mathbb{C}\setminus\Omega_{\Gamma})\cap J_{\zeta_{1},\zeta_{2}}=J_{\zeta_{0},\zeta_{2}}$. Similarly, $\sigma(T|_{\mathcal{M}^{\prime}})\subset J_{\zeta_{1},\zeta_{0}}$.∎

Remark 2.8**.**

Let $T$ be an a.c. unitary operator, and let $\theta$ be an inner function. Then $\theta(T)$ is a unitary operator, in particular, $\theta(T)$ is invertible. In the next section, examples of nonunitary operators satisfying the assumptions of Theorem 2.5 are given (Example 3.5).

Remark 2.9**.**

As it will turn out in the beginning of the next section, operators satisfying the assumptions of Theorem 2.5 cannot be quasianalytic. But if one replaces the assumption “$\mu(\sigma(T)\cap\mathbb{T})>0$” of Theorem 2.5 by “$\operatorname{supp}\mu\cap\sigma(T)\neq\emptyset$”, where $\operatorname{supp}\mu$ is the closed support of $\mu$, then the operator satisfying these modified assumptions can be quasianalytic, see Example 3.4 below.

3. Quasianalyticity and invertibility of inner functions of operators

For the definition of quasianalyticity we refer to [K16] or [K01] and to [E], [K01], [KS14], [KS15], [K16], [Gam] for examples of quasianalytic operators. We recall only that the set $\pi(T)$ of quasianalyticity of an operator $T$ is a Borel set, $\pi(T)\subset\sigma(T)\cap\mathbb{T}$, and if $T$ is a quasianalytic operator, then $\pi(T)\neq\emptyset$. Furthermore, $\pi(T)\subset\sigma(T|_{\mathcal{M}})$ for every nonzero $\mathcal{M}\in\operatorname{Lat}T$ for an a.c. polynomially bounded operator $T$ [K16, Proposition 35]. Therefore, operators satisfying the assumptions of Theorem 2.5 cannot be quasianalytic. In this section, examples of quasianalytic operators $Q$ are given such that $\theta_{\mu}(Q)$ is invertible with $\mu(\sigma(Q)\cap\mathbb{T})>0$ for a pure atomic measure $\mu$, and also $B(Q)$ is invertible for a Blaschke product $B$ with $\{\zeta\in\mathbb{T}\ :$ $\ \zeta$ is a nontangential limit of zeros of $B\}\subset\sigma(Q)$.

We need the following simple lemma. Recall that for $\lambda\in\mathbb{D}$ the set $D(\lambda)$ is defined in (1.2).

Lemma 3.1**.**

Let $\{\zeta_{n}\}_{n\in\mathbb{N}}$ be a sequence of points from $\mathbb{T}$ such that $\zeta_{n}\neq\zeta_{l}$ for $n\neq l$. Then there exist $\{r_{n}\}_{n\in\mathbb{N}}$, $\{a_{n}\}_{n\in\mathbb{N}}$ and $\Lambda\subset\mathbb{D}$ such that $0<r_{n}<1$, $a_{n}>0$, $\sum_{n\in\mathbb{N}}(1-r_{n})<\infty$, $\sum_{n\in\mathbb{N}}a_{n}<\infty$,

[TABLE]

is a simply connected domain, $B:=\prod_{\lambda\in\Lambda}b_{\lambda}$ is an interpolating Blaschke product, $\zeta_{n}$ is a nontangential limit of points from $\Lambda$ for every $n\in\mathbb{N}$,

[TABLE]

Proof.

The domain $\mathcal{G}$ is constructed by induction. Take $\{r_{1,n}\}_{n\in\mathbb{N}}$ such that $0<r_{1,n}<1$ and $\sum_{n\in\mathbb{N}}(1-r_{1,n})<\infty$. Take $r_{1}\geq r_{1,1}$. Since $\zeta_{2}\not\in\operatorname{clos}D(r_{1}\zeta_{1})$, there exists $r_{2}\geq r_{1,2}$ such that

[TABLE]

Since

[TABLE]

there exists $r_{3}\geq r_{1,3}$ such that

[TABLE]

and so on.

After $\{r_{n}\}_{n\in\mathbb{N}}$ has been constructed, take $\{a_{n}\}_{n\in\mathbb{N}}$ such that $a_{n}>0$ and

[TABLE]

By Lemma 1.1, $\theta_{\mu}$ with $\mu=\sum_{n\in\mathbb{N}}a_{n}\delta_{\zeta_{n}}$ satisfies the conclusion of the lemma.

Before constructing an interpolating Blaschke product we recall the definition and the needed facts. Let $\Lambda\subset\mathbb{D}$ be such that $\sum_{\lambda\in\Lambda}(1-|\lambda|)<\infty$. Set $B_{\Lambda}=\prod_{\lambda\in\Lambda}b_{\lambda}$. Then $B_{\Lambda}$ is a Blaschke product with simple zeros. $B_{\Lambda}$ is called interpolating, if for every $\alpha\in\ell^{\infty}$ there exists $\varphi\in H^{\infty}$ such that $\varphi|_{\Lambda}=\alpha$. It is known that $B_{\Lambda}$ is an interpolating Blaschke product if and only if there exists $c>0$ such that

[TABLE]

(see, for example, [N, Secs. C.3.2.15–C.3.2.18]). It follows exactly from the definition that if $B_{\Lambda}$ is an interpolating Blaschke product, then $B_{\Lambda^{\prime}}$ is an interpolating Blaschke product, too, for every $\Lambda^{\prime}\subset\Lambda$.

Let $\Lambda\subset\cup_{n\in\mathbb{N}}[r_{n}\zeta_{n},\zeta_{n})$ be such that $B_{\Lambda}$ is an interpolating Blaschke product. It follows from Lemma 1.3, (3.1) and the definition of $\mathcal{G}$ that $\inf_{\mathcal{G}}|B_{\Lambda}|\geq\frac{c}{3}$.

To construct Blaschke product, take $\{\rho_{n}\}_{n\in\mathbb{N}}$ such that $0<\rho_{n}<1$ and $\sup_{n\in\mathbb{N}}\frac{1-\rho_{n+1}}{1-\rho_{n}}<1.$ Then $\prod_{n\in\mathbb{N}}b_{\rho_{n}\xi_{n}}$ is an interpolating Blaschke product for arbitrary $\{\xi_{n}\}_{n\in\mathbb{N}}\subset\mathbb{T}$, see, for example, [D, Theorem 9.2] or [N, Sec. C.3.3.4(c)]. Therefore, $\prod_{n\in\mathfrak{N}}b_{\rho_{n}\xi_{n}}$ is an interpolating Blaschke product for every $\mathfrak{N}\subset\mathbb{N}$.

We can consider the partition into disjoint classes

[TABLE]

where $\rho_{n,k}\to 1$ when $k\to\infty$ for every $n\in\mathbb{N}$. Set

[TABLE]

Then $B$ satisfies the conclusion of the lemma. ∎

Lemma 3.2**.**

Suppose that $T$ is an a.c. polynomially bounded operator, $\mathcal{G}\subset\mathbb{D}$ is a simply connected domain, $g\colon\mathbb{D}\to\mathcal{G}$ is a conformal mapping, $Q=g(T)$, $\psi\in H^{\infty}$, and $\inf_{\mathcal{G}}|\psi|>0$. Then $Q$ is an a.c. polynomially bounded operator and $\psi(Q)$ is invertible.

Proof.

$Q$ is an a.c. polynomially bounded operator and $\psi(Q)=(\psi\circ g)(T)$ by [K15, Proposition 2]. Since $1/\psi\circ g\in H^{\infty}$, it is easily checked that $\psi(Q)^{-1}=(1/\psi\circ g)(T)$. ∎

Theorem 3.3**.**

Let $\{\zeta_{n}\}_{n\in\mathbb{N}}$ be a sequence of points from $\mathbb{T}$ such that $\zeta_{n}\neq\zeta_{l}$ for $n\neq l$. Then there exist $\{a_{n}\}_{n\in\mathbb{N}}$, $\Lambda\subset\mathbb{D}$, and a quasianalytic contraction $Q$ such that $a_{n}>0$, $\sum_{n\in\mathbb{N}}a_{n}<\infty$, $B:=\prod_{\lambda\in\Lambda}b_{\lambda}$ is an interpolating Blaschke product, $\zeta_{n}$ is a nontangential limit of points from $\Lambda$ for every $n\in\mathbb{N}$, $\cup_{n\in\mathbb{N}}\{\zeta_{n}\}\subset\sigma(Q)$,

[TABLE]

Proof.

Let $\mathcal{G}$, $\{a_{n}\}_{n\in\mathbb{N}}$, $\Lambda\subset\mathbb{D}$ be constructed in Lemma 3.1. Let $g\colon\mathbb{D}\to\mathcal{G}$ be a conformal mapping. Although $\partial\mathcal{G}$ is not a Jordan curve, $\partial\mathcal{G}$ is a rectifiable curve, since $\sum_{n\in\mathbb{N}}(1-r_{n})<\infty$. Therefore, $g^{\prime}\in H^{1}$, see [Po75, Theorem 10.11] and [Po92, Sec. 6.2,6.3]. In particular, $\sum_{k=0}^{\infty}|\widehat{g}(k)|<\infty$ by Hardy’s inequality, see, for example, [D, Corollary of Theorem 3.15]. Therefore, $g$ is continuous on $\operatorname{clos}\mathbb{D}$. Thus, $g$ belongs to the disc algebra.

Let $T$ be a quasianalytic contraction such that $m(g(\pi(T))\cap\mathbb{T})>0$. Set $Q=g(T)$. Since $g^{\prime}\in H^{1}$, $Q$ is quasianalytic and $\pi(Q)=g(\pi(T))\cap\mathbb{T}$ by [K15, Remark 8 and Corollary 13]. If, for example, $\pi(T)=\mathbb{T}$, then $\pi(Q)=\mathbb{T}$.

By Lemmas 3.1 and 3.2, $\theta_{\mu}(Q)$ with $\mu=\sum_{n\in\mathbb{N}}a_{n}\delta_{\zeta_{n}}$ and $B(Q)$ are invertible.

By the spectral mapping theorem for functions from the disc algebra and polynomially bounded operators, $\sigma(Q)=g(\sigma(T))$, see Remark 2.4 for references. Therefore, if $g^{-1}(\zeta_{n})\cap\sigma(T)\neq\emptyset$ for some $n\in\mathbb{N}$, then $\zeta_{n}\in\sigma(Q)$. (The set $g^{-1}(\zeta_{n})$ consists of two points from $\mathbb{T}$ for every $n\in\mathbb{N}$.) If, for example, $\sigma(T)=\operatorname{clos}\mathbb{D}$, then $\sigma(Q)=\operatorname{clos}\mathcal{G}$.∎

In the following examples, a construction similar to the construction from Theorem 3.3 is used.

Example 3.4**.**

Suppose that $E\subset\mathbb{T}$ is a set satisfying the assumptions of Lemma 1.2 and $E$ contains no isolated points. Suppose that $\mu$ is a finite positive Borel measure on $E$ such that $\mu(\{\zeta\})=0$ for every $\zeta\in E$, and $\operatorname{supp}\mu=E$ (where $\operatorname{supp}\mu$ is the closed support of $\mu$). We have $\mathbb{T}\setminus E=\cup_{n\in\mathbb{N}}\mathcal{J}_{\zeta_{1n},\zeta_{2n}}$. Take $1/2<r<1$. Relabeling $\{\mathcal{J}_{\zeta_{1n},\zeta_{2n}}\}_{n\in\mathbb{N}}$ if necessarily, we find $N\in\mathbb{N}$ such that $\operatorname{clos}D(r\zeta_{1n})\cap\operatorname{clos}D(r\zeta_{2n})\neq\emptyset$ for every $n\geq N$ and $\operatorname{clos}D(r\zeta_{1n})\cap\operatorname{clos}D(r\zeta_{2n})=\emptyset$ for every $n<N$. Let $n\geq N$. Then

[TABLE]

where $\mathcal{G}_{1n}$, $\mathcal{G}_{2n}$ are disjoint simply connected domains, $\partial\mathcal{G}_{1n}$, $\partial\mathcal{G}_{2n}$ are rectifiable Jordan curves,

[TABLE]

and $\partial\mathcal{G}_{2n}\cap\mathbb{T}=\mathbb{T}\setminus\mathcal{J}_{\zeta_{1n},\zeta_{2n}}$. It is easy to see that

[TABLE]

For $n<N$, it is easy to construct a simply connected domain $\mathcal{G}_{1n}$ satisfying (3.3) and (3.2) and such that $\partial\mathcal{G}_{1n}$ is a rectifiable Jordan curve. For example, let

[TABLE]

For every $n\in\mathbb{N}$, let $g_{n}\colon\mathbb{D}\to\mathcal{G}_{1n}$ be a conformal mapping. Since $\partial\mathcal{G}_{1n}$ is a rectifiable curve, we have $g_{n}^{\prime}\in H^{1}$, therefore, $g_{n}$ belongs to the disc algebra (see the proof of Theorem 3.3 for references).

Let $n\in\mathbb{N}$. Let $T$ be a quasianalytic contraction such that

[TABLE]

Set $Q=g_{n}(T)$. Then $Q$ is quasianalytic and $\pi(Q)=g_{n}(\pi(T))\cap\mathbb{T}$. If, for example, $\pi(T)=\mathbb{T}$, then $\pi(Q)=\mathcal{J}_{\zeta_{1n},\zeta_{2n}}$. (See the proof of Theorem 3.3 for references.) Since $\pi(Q)\subset\sigma(Q)$ (see the references in the beginning of this section), we conclude that $\zeta_{1n}$, $\zeta_{2n}\in\sigma(Q)\cap E$. By (3.3), Lemmas 1.2 and 3.2, $\theta_{\mu}(Q)$ is invertible. Thus, $Q$ is an example to Remark 2.9.

Example 3.5**.**

Let $E$ and $\mu$ be as in Example 3.4, and let $\{\mathcal{G}_{1n}\}_{n\in\mathbb{N}}$ and $\{g_{n}\}_{n\in\mathbb{N}}$ be constructed in Example 3.4. Let $\{T_{n}\}_{n\in\mathbb{N}}$ be a family of a.c. uniformly polynomially bounded operators, in particular, a.c. contractions. For every $n\in\mathbb{N}$, set $Q_{n}=g_{n}(T_{n})$. By (2.8) applied to $T_{n}$ and $g_{n}$, taking into account that $g_{n}$ is from the disc algebra, we conclude that $\sigma(Q_{n})=g_{n}(\sigma(T_{n}))$. Furthermore, $(Q_{n}-a)^{-1}=(1/(g_{n}-a))(T_{n})$ for $a\not\in\operatorname{clos}\mathcal{G}_{1n}$ and $\theta_{\mu}(Q_{n})^{-1}=(1/\theta_{\mu}\circ g_{n})(T_{n})$ (see the proof of Lemma 3.2). By (3.3) and Lemma 1.2, $\sup_{n\in\mathbb{N}}\|\theta_{\mu}(Q_{n})^{-1}\|<\infty$.

Set

[TABLE]

Then $Q$ is an a.c. polynomially bounded operator, $\theta_{\mu}(Q)$ is invertible, and

[TABLE]

If $\sigma(T_{n})=\operatorname{clos}\mathbb{D}$ for all $n\in\mathbb{N}$, then $\sigma(Q)=\operatorname{clos}\mathcal{G}$. Thus, $Q$ is an example to Remark 2.8.

4. Invertibility of functions of weighted shifts

Recall the definition of a bilateral weighted shift, see [S]. Let $\omega\colon\mathbb{Z}\to(0,\infty)$ be a function such that $\sup_{n\in\mathbb{Z}}\frac{\omega(n+1)}{\omega(n)}<\infty$. Set

[TABLE]

The bilateral weighted shift $S_{\omega}\in\mathcal{L}(\ell^{2}_{\omega})$ acts according to the formula

[TABLE]

If $\omega(n)=1$ for every $n\in\mathbb{Z}$, then $S_{\omega}$ is called the simple bilateral shift. Clearly, the simple bilateral shift is a unitary operator, and $S_{\omega}$ is similar to the simple bilateral shift if and only if

[TABLE]

(see [S, Theorem $2^{\prime}$]). Easy computation shows (see [S, Corollary of Proposition 7]) that

[TABLE]

Recall that $\sigma(S_{\omega})$ is a closed disc or a (may be degenerate) annulus centered at origin [S, Theorem 5]. By [S, Corollary of Theorem 2], if $S_{\omega}$ is a power bounded weighted shift ($\sup_{k\in\mathbb{N}}\|S_{\omega}^{k}\|<\infty$), then $S_{\omega}$ is similar to a contractive weighted shift, which is necessarily a.c.. Therefore, $\varphi(S_{\omega})$ is well defined for a power bounded weighted shift $S_{\omega}$ and $\varphi\in H^{\infty}$. In this section it is proved that if $\liminf|\varphi(z)|=0$ when $|z|\to 1$, $S_{\omega}$ is a power bounded weighted shift with $\mathbb{T}\subset\sigma(S_{\omega})$, and $\varphi(S_{\omega})$ is invertible, then $S_{\omega}$ is similar to the simple bilateral shift.

We need the following simple lemmas.

Lemma 4.1**.**

Suppose that $\{\alpha_{n}\}_{n=0}^{\infty}$, $\{\beta_{n}\}_{n=0}^{\infty}$ are sequences of non-negative numbers, $C>0$, $\{r_{j}\}_{j}\subset(0,1)$, $r_{j}\to 1$, and

[TABLE]

Then $\limsup_{n}(C\beta_{n}-\alpha_{n})\geq 0$.

Proof.

Suppose that there exist $\varepsilon>0$ and an index $N$ such that $C\beta_{n}-\alpha_{n}<-\varepsilon$ for every $n\geq N+1$. Then

[TABLE]

The right side of this inequality tends to $-\infty$ when $r_{j}\to 1$, a contradiction.∎

Lemma 4.2**.**

Suppose that $\omega\colon\mathbb{Z}\to(0,\infty)$ is such that

[TABLE]

$C>0$, $\{r_{j}\}_{j}\subset(0,1)$, $r_{j}\to 1$, and

[TABLE]

Then $\omega$ satisfies (4.1).

Proof.

First, note that (4.3) implies that $\sup_{n\geq 0}\omega(n)<\infty$. It follows from (4.4) and Lemma 4.1 that there exist a subsequence $\{n_{j}\}_{j}$ and a sequence $\{\varepsilon_{j}\}_{j}\subset(0,\infty)$ such that $n_{j}\to+\infty$, $\varepsilon_{j}\to 0$, and

[TABLE]

In particular, $\sup_{j}\omega(-n_{j}-1)<\infty$. This relation and (4.3) imply that $\sup_{n\geq 0}\omega(-n)<\infty$.

If $\inf_{n\geq 0}\omega(-n)=0$, then (4.3) implies that $\omega(n)=0$ for every $n\in\mathbb{Z}$. If $\inf_{n\geq 0}\omega(n)=0$, then (4.3) implies that $\omega(n)\to 0$ when $n\to+\infty$. In particular, $\omega(n_{j})\to 0$. Then, by (4.5), $\omega(-n_{j}-1)\to 0$. As was mentioned above, this relation with (4.3) implies that $\omega(n)=0$ for every $n\in\mathbb{Z}$. Thus, $\inf_{n\in\mathbb{Z}}\omega(n)>0$.∎

Theorem 4.3**.**

Suppose that $S_{\omega}$ is a power bounded weighted shift, $\mathbb{T}\subset\sigma(S_{\omega})$, $\varphi\in H^{\infty}$, $\liminf|\varphi(z)|=0$ when $|z|\to 1$, and $\varphi(S_{\omega})$ is invertible. Then $S_{\omega}$ is similar to the simple bilateral shift.

Proof.

As was mentioned above, $S_{\omega}$ is similar to an a.c. contraction ([S, Corollary of Theorem 2]); thus, $\varphi(S_{\omega})$ is well defined.

Let $0<c<1/\|\varphi(S_{\omega})^{-1}\|$. By assumption, there exists $\{\lambda_{j}\}_{j}\subset\mathbb{D}$ such that $|\lambda_{j}|\to 1$ and $|\varphi(\lambda_{j})|\leq c$. By Lemma 2.3, $\lambda_{j}\not\in\sigma(S_{\omega})$ for all $j$. Since $\mathbb{T}\subset\sigma(S_{\omega})\subset\operatorname{clos}\mathbb{D}$ (the latter inclusion is due to power boundedness of $S_{\omega}$), and $\sigma(S_{\omega})$ is an annulus ([S, Theorem 5]), we conclude that $\sigma(S_{\omega})=\mathbb{T}$.

Let $u_{n}\in\ell^{2}_{\omega}$, $u_{n}(k)=0$, $n\neq k$, $u_{n}(n)=1$ ($n\in\mathbb{Z}$). Then

[TABLE]

for every $\lambda\in\mathbb{D}$. By Lemma 2.3, there exists $C>0$ such that

[TABLE]

for every $u\in\ell^{2}_{\omega}$ and every $\lambda\in\mathbb{D}$ such that $|\varphi(\lambda)|\leq c$. We obtain that

[TABLE]

with $|\lambda_{j}|\to 1$. Since $S_{\omega}$ is power bounded, $\omega$ satisfies (4.3) (see (4.2)). Applying Lemma 4.2 with $r_{j}=|\lambda_{j}|^{2}$ we obtain (4.1). As was mentioned in the beginning of this section, (4.1) implies the similarity of $S_{\omega}$ to the simple bilateral shift. ∎

Remark 4.4**.**

Recall that for every singular inner function $\theta$ there exists a weight $\omega$ such that $\omega(n)=1$ for $n\geq 0$, the weighted shift $S_{\omega}$ is a quasianalytic contraction, $\sigma(S_{\omega})=\mathbb{T}$, and $\operatorname{clos}\theta(S_{\omega})\ell^{2}_{\omega}=\ell^{2}_{\omega}$ [E, Theorem 5.9].

The author is grateful to the referee for careful reading of the paper and correcting many inaccuracies and misprints.