Mean ergodic theorems for power bounded measures


Mustafayev H. , Şevli H.

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, vol.500, no.1, 2021 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 500 Issue: 1
  • Publication Date: 2021
  • Doi Number: 10.1016/j.jmaa.2021.125090
  • Title of Journal : JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

Abstract

Let G be a locally compact abelian group and let M(G) be the convolution measure algebra of G. A measure mu is an element of M(G) is said to be power bounded if sup(n >= 0) parallel to mu(n)parallel to(1) < infinity, where mu(n) denotes nth convolution power of mu. We show that if mu is an element of M(G) is power bounded and A = [a(n,k)](n,k=0)(infinity) is a strongly regular matrix, then the limit lim(n ->infinity) Sigma(infinity)(k=0) a(n,k) mu(k) exists in the weak* topology of M(G) and is equal to the idempotent measure theta, where (theta) over cap = 1(int)F(mu). Here, (theta) over cap is the Fourier-Stieltjes transform of theta, F-mu :={gamma is an element of Gamma : (mu) over cap(gamma) = 1}, and 1(int) F-mu is the characteristic function of int F-mu. Some applications are also given. (C) 2021 Elsevier Inc. All rights reserved.