The reflexivity of hyperexpansions and their Cauchy dual operators

TL;DR

This paper investigates the reflexivity properties of hyperexpansions and their Cauchy duals, establishing reflexivity results for cyclic hyperexpansive operators and certain Bergman-type operators, advancing understanding in operator theory.

Contribution

It proves reflexivity of cyclic completely hyperexpansive operators and the Cauchy dual of 2-hyperexpansive operators, including Bergman-type operators, which was previously unknown.

Findings

Cyclic completely hyperexpansive operators are reflexive.

The Cauchy dual of any 2-hyperexpansive operator is reflexive.

Bergman-type operators are reflexive.

Abstract

We discuss the reflexivity of hyperexpansions and their Cauchy dual operators. In particular, we show that any cyclic completely hyperexpansive operator is reflexive. We also establish the reflexivity of the Cauchy dual of an arbitrary $2$-hyperexpansive operator. As a consequence, we deduce the reflexivity of the so-called Bergman-type operator, that is, a left-invertible operator $T$ satisfying the inequality $TT^* + (T^*T)^{-1} \leqslant 2 I_{\mathcal H}.$