Some convergence theorems in Banach algebras

J. J. Koliha

Vol. 52 (1974), No. 2, 467–473

DOI: 10.2140/pjm.1974.52.467

Abstract

This paper is concerned with finding necessary and sufficient conditions for the convergence of the sequence {f_n(a)} of elements of Banach algebra, where {f_n} is a sequence of analytic functions imitating the behavior of the sequence of integral powers. In particular, it is shown that the sequence {aⁿ} converges iff the spectrum of a (with the possible exception of the point λ = 1) lies in the open unit disc and λ = 1 is a pole of (λ − a)⁻¹ of order ≦ 1.

Mathematical Subject Classification 2000

Primary: 46H05

Milestones

Received: 7 August 1973

Revised: 28 November 1973

Published: 1 June 1974