An Exposition on Weak Stability of Operators

TL;DR

This paper provides a comprehensive survey on the weak stability of bounded linear operators, highlighting recent results, open questions, and research trends, with a focus on Hilbert space operators and unitary operators.

Contribution

It offers an updated review of sixty years of literature on weak operator stability, including new results and insights into the prominence of Hilbert spaces and unitary operators.

Findings

Majority of research focuses on Hilbert-space operators

Weak stability of unitary operators is central to the problem

Includes some new results and unresolved questions

Abstract

This is an expository-survey on weak stability of bounded linear operators acting on normed spaces in general and, in particular, on Hilbert spaces. The paper gives a comprehensive account of the problem of weak operator stability, containing a few new results and some unanswered questions. It also gives an updated review of the literature on the weak stability of operators over the past sixty years, including present-day research trends. It is verified that the majority of the weak stability literature is concentrated on Hilbert-space operators. We discuss why this preference occurs and also why the weak stability of unitary operators is central to the Hilbert-space stability problem.