The Kitai Criterion and backward shifts

TL;DR

This paper proves that in any separable infinite-dimensional Banach space, there exists a bounded linear operator satisfying the Kitai Criterion, using quasisimilarity and properties of backward weighted shifts.

Contribution

It introduces a method to construct operators satisfying the Kitai Criterion on any separable infinite-dimensional Banach space.

Findings

Existence of operators satisfying the Kitai Criterion in all such spaces

Use of quasisimilarity to establish the result

Identification of backward weighted shifts as key examples

Abstract

It is proved that for any separable infinite dimensional Banach space $X$, there is a bounded linear operator $T$ on $X$ such that $T$ satisfies the Kitai Criterion. The proof is based on quasisimilarity argument and on showing that $I+T$ satisfies the Kitai Criterion for certain backward weighted shifts $T$.