Characterisation of Ces\`aro and $L$-Asymptotic Limits of Matrices

TL;DR

This paper characterizes all Cesàro and L-asymptotic limits of powerbounded matrices, showing their equivalence and exploring their properties, including norm bounds and connections to invertible matrices with unit vectors.

Contribution

It provides a complete characterization of Cesàro and L-asymptotic limits for powerbounded matrices and establishes their equivalence, extending understanding of their behavior and properties.

Findings

Cesàro and L-asymptotic limits coincide for all powerbounded matrices.

The norm of the L-asymptotic limit is ≥ 1 unless it is zero.

The same norm bound applies to Cesàro limits of non-powerbounded operators.

Abstract

The main goal of this paper is to characterise all the possible Ces\`aro and $L$-asymptotic limits of powerbounded, complex matrices. The investigation of the $L$-asymptotic limit of a powerbounded operator goes back to Sz.-Nagy and it shows how the orbit of a vector behaves with respect to the powers. It turns out that the two types of asymptotic limits coincide for every powerbounded matrix and a special case is connected to the description of the products $SS^*$ where $S$ runs through those invertible matrices which have unit columnvectors. We also show that for any powerbounded operator acting on an arbitrary complex Hilbert space the norm of the $L$-asymptotic limit is greater than or equal to 1, unless it is zero; moreover, the same is true for the Ces\`aro asymptotic limit of a not necessarily powerbounded operator, if it exists.