Asymptotic limits, Banach limits, and Cesàro means

Abstract

Every new inner product in a Hilbert space is obtained from the original one by means of a unique positive operator$.$ The first part of the paper is a survey on applications of such a technique, including a characterization of similarity to isometries$.$ The second part focuses on Banach limits for dealing with power bounded operators. It is shown that if a power bounded operator for which the sequence of shifted Cesàro means converges (at least in the weak topology) uniformly in the shift parameter, then it has a Cesàro asymptotic limit coinciding with its $\phi$-asymptotic limit for all Banach limits $\phi$.