On the power-bounded operators of classes $C_{0 \cdot}$ and $C_{1 \cdot}$

TL;DR

This paper extends characterizations of stability and non-vanishing properties from contractions to power-bounded operators using bounded backward sequences, revealing new duality relations with the adjoint operator.

Contribution

It generalizes previous results on contractions to the broader class of power-bounded operators, establishing criteria based on backward sequences and adjoint properties.

Findings

A power-bounded operator is strongly stable if and only if its adjoint lacks nonzero bounded backward sequences.

A power-bounded operator is non-vanishing if and only if its adjoint has many bounded backward sequences.

The results unify stability concepts for contractions and power-bounded operators through backward sequence analysis.

Abstract

By a bounded backward sequence of the operator $T$ we mean a bounded sequence $\{x_n\}$ satisfying $Tx_{n+1}=x_n$. In \cite{Pa} we have characterized contractions with strongly stable nonunitary part in terms of bounded backward sequences. The main purpose of this work is to extend that result to power-bounded operators. Aditionally, we show that a power-bounded operator is strongly stable ($C_{0 \cdot} $) if and only if its adjoint does not have any nonzero bounded backward sequence. Similarly, a power-bounded operator is non-vanishing ($C_{1 \cdot} $) if and only if its adjoint has a lot of bounded backward sequences.