Some convergence theorems in Banach algebras
J. J. Koliha · Pacific Journal of Mathematics · 1974
This paper is concerned with finding necessary and sufficient conditions for the convergence of the sequence {/"(&)} of elements of Banach algebra, where {/J is a sequence of analytic functions imitating the behavior of the sequence of integral powers.In particular, it is shown that the sequence {a 71 } 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 (λa)' 1 of order ^ 1.4. J. J. Koliha, Ergodic theory and averaging iterations, Canad.J. Math., 25 (1973), 14-23. 5., The solution of linear equations in normed spaces by averaging iteration,