On power bounded operators with holomorphic eigenvectors, II

In [U] (among other results), M. Uchiyama gave the necessary and sufficient conditions for contractions to be similar to the unilateral shift $S$ of multiplicity $1$ in terms of norm-estimates of complete analytic families of eigenvectors of their adjoints. In [G2], it was shown that this result for contractions can't be extended to power bounded operators. Namely, a cyclic power bounded operator was constructed which has the requested norm-estimates, is a quasiaffine transform of $S$, but is not quasisimilar to $S$. In this paper, it is shown that the additional assumption on a power bounded operator to be quasisimilar to $S$ (with the requested norm-estimates) does not imply similarity to $S$. A question whether the criterion for contractions to be similar to $S$ can be generalized to polynomially bounded operators remains open. Also, for every cardinal number $2\leq N\leq \infty$ a power bounded operator $T$ is constructed such that $T$ is a quasiaffine transform of $S$ and $\dim\ker T^*=N$. This is impossible for polynomially bounded operators. Moreover, the constructed operators $T$ have the requested norm-estimates of complete analytic families of eigenvectors of $T^*$

On expansive operators that are quasisimilar to the unilateral shift of finite multiplicity

An operator $T$ on a Hilbert space $\mathcal H$ is called expansive, if $\|Tx\|\geq \|x\|$ ($x\in\mathcal H$). It is proves that if an expansive operator $T$ is quasisimilar to the unilateral shift of finite multiplicity $N\geq 2$, then $I-T^*T$ is of trace class and there exist invariant subspaces $\mathcal M_j$ ($j=1,\ldots, N$) of $T$ such that the restriction $T|_{\mathcal M_j}$ of $T$ on $\mathcal M_j$ is similar to the unilateral shift of multiplicity $1$ for every $j=1,\ldots, N$, and $\mathcal H=\vee_{j=1}^N\mathcal M_j$. If an expansive operator $T$ is quasisimilar to the unilateral shift of multiplicity $1$, then $I-T^*T$ is of trace class and there exist invariant subspaces $\mathcal M_1$ and $\mathcal M_2$ of $T$ such that the restriction $T|_{\mathcal M_j}$ of $T$ on $\mathcal M_j$ is similar to the unilateral shift of multiplicity $1$ for $j=1,2$, and $\mathcal H=\mathcal M_1\vee\mathcal M_2$

Example of quasianalytic contraction whose spectrum is a proper subarc of the unit circle

A partial answer on [KS2, Question 2] is given. Namely, an operator $R$ similar to a quasianalytic contraction whose quasianalytic spectral set is equal to its spectrum and is a proper subarc of the unit circle is constructed, but no estimates of $\|R^{-1}\|$ is given

On existence of shift-type invariant subspaces for polynomially bounded operator

A particular case of results from [K2] is as follows. Let the unitary asymptote of a contraction $T$ contain the bilateral shift (of finite or infinite multiplicity). Then there exists an invariant subspace $\mathcal M$ of $T$ such that $T|_{\mathcal M}$ is similar to the unilateral shift of the same multiplicity. The proof is based on the Sz.-Nagy--Foias functional model for contractions. In the present paper this result is generalized to polynomially bounded operators, but in the simplest case. Namely, it is proved that if the unitary asymptote of a polynomially bounded operator $T$ contains the bilateral shift of multiplicity $1$, then there exists an invariant subspace $\mathcal M$ of $T$ such that $T|_{\mathcal M}$ is similar to the unilateral shift of multiplicity $1$. The proof is based on a result from [B].Comment: The estimates of the norms of intertwining transformations in terms of the polynomial bound of the operator under consideration are give

Invertibility of functions of operators and existence of hyperinvariant subspaces

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.Comment: Many inaccuracies and misprints are corrected by the refere