Characterization of Cesaro- and L-asymptotic limits of matrices

Gehér, György (2014) Characterization of Cesaro- and L-asymptotic limits of matrices. LINEAR AND MULTILINEAR ALGEBRA, inpres. inpress. ISSN 0308-1087

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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.

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