# On power bounded operators with holomorphic eigenvectors, II

## TL;DR

This paper investigates the limitations of extending criteria for similarity to the unilateral shift from contractions to power bounded operators, providing new constructions and highlighting open questions in operator theory.

## Contribution

It demonstrates that being quasisimilar to the unilateral shift with certain eigenvector norm-estimates does not imply similarity for power bounded operators, and constructs operators with specific kernel dimensions.

## Key findings

- Power bounded operators with eigenvector estimates are not necessarily similar to the shift.
- Constructed operators have prescribed kernel dimensions, unlike polynomially bounded operators.
- The extension of contraction criteria to polynomially bounded operators remains unresolved.

## Abstract

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 $T^*$.