Asymptotic limits, Banach limits, and Ces\`aro means

TL;DR

This paper surveys the use of positive operators in Hilbert spaces to characterize similarity to isometries and explores Banach limits in relation to power bounded operators, Cesàro means, and asymptotic limits.

Contribution

It provides a comprehensive survey of inner product transformations and establishes new results on Cesàro asymptotic limits for power bounded operators using Banach limits.

Findings

Characterization of similarity to isometries via positive operators.

Conditions under which Cesàro means converge uniformly in shift parameter.

Equivalence of Cesàro and symptotic limits for certain operators.

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 Ces\`aro means converges (at least in the weak topology) uniformly in the shift parameter, then it has a Ces\`aro asymptotic limit coinciding with its $\varphi$-asymptotic limit for all Banach limits $\varphi$.