Characterization of Cesàro and L-asymptotic limits of matrices

Abstract

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.

Keywords:

• L-asymptotic limit
• Cesàro asymptotic limit
• power-bounded matrix
• positive semidefinite matrix

AMS Subject Classifications:

• 47A45
• 47B65

Acknowledgements

The author emphasizes his thank to L. Kérchy for his supervision. He also thanks to L. Ozsvárt for a useful suggestion. This research was realized in the frames of TÁMOP 4.2.4. A/2-11-1-2012-0001 ‘National Excellence Program – Elaborating and operating an inland student and researcher personal support system’. The project was subsidised by the European Union and co-financed by the European Social Fund. The author was also supported by the ‘Lendület’ Program (LP2012-46/2012) of the Hungarian Academy of Sciences.