Characterization of Cesàro andL-asymptotic limits of matrices
György Pál Gehér · Linear and Multilinear Algebra · 2014
The main goal of this paper is to characterize all the possible Cesàro and -asymptotic limits of (self-adjoint iterates of) power bounded, complex matrices. The investigation of the -asymptotic limit of a power-bounded operator goes back to Sz.-Nagy and it shows how the orbit of a vector behaves with respect to the powers. In this paper, we prove that the two types of asymptotic limits coincide for every power-bounded matrix and a special case is connected to the description of the products , where runs through those invertible matrices which have unit column vectors. We also show that for any power-bounded operator acting on an arbitrary complex Hilbert space the norm of the -asymptotic limit is greater than or equal to 1, unless it is zero; moreover, the same is true for the Cesàro asymptotic limit of a not necessarily power-bounded operator, if it exists.