Some inequalities on power bounded operators acting on locally convex spaces

Abstract

In this paper are presented some evaluations regarding characterizations of power boundedness of an operator from B P (X) algebra. This represents a generalization from the Banach spaces framework to the locally convex spaces where the operators acting on them are universally bounded. The authors extend some Ritt type theorems according to Kreiss conditions in the context of B P (X) algebra.

On the Ritt condition on Locally Convex Vector Spaces

“…In the following theorem, our goal is to get an upper bound for \| T \| \scrP under condition (2.3) in the case of universally bounded operators acting on a locally convex space X. Related to this, F. Pater [13] showed that if condition (2.3) holds, then \| T n \| \scrP = O(n). Proof.…”

“…In [10], it was shown that if Ritt resolvent condition holds for an operator T acting on a Banach space, then \| T n \| =| O(\mathrm{ \mathrm{ \mathrm{ n) as n -\rightar \infty , and \bigm\| \bigm\| T n -T n+1 \bigm\| \bigm\| \rightar 0 as n -\rightar \infty . This result was generalized by Pater for operators acting on locally convex spaces [13,Theorem 3]. The first aim of the present article is to improve this result by giving a characterization of the Ritt resolvent condition by two geometric properties of the powers.…”

“…This characterization in the case of bounded linear operators acting on Banach spaces was presented in [2]. In [13]…”