An Exposition on Weak Stability of Operators

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.