On a characterization of infinite complex matrices mapping the space of analytic sequences into itself

Louise Arakelian Raphael

Vol. 27 (1968), No. 1, 123–126

DOI: 10.2140/pjm.1968.27.123

Abstract

Let S be the space of all complex sequences. An element u = {u_n}_n=0^∞ of S is called analytic if for some constant M > 0, |u_n|≦ Mⁿ⁺¹ for n = 0,1,2,⋯ . By A denote the space of all analytic sequences. Clearly A is the space of all complex functions analytic at zero. I. Heller has proved

Theorem 1. The transformation y_n = ∑ _m=0^∞c_nmu_m maps A into A if and only if for every p > 0 there exists a q > 0 and a constant M > 0 such that |c_nm|≦ Mp^m∕qⁿ for m,n = 0,1,2,⋯ ; and also if and only if the function G of two complex variables (i.e., in E × E, where E is the complex plane) respresented by the double power series G(z,y) = ∑ _m,n=0^∞c_nmz^myⁿ be regular on E × 0.

The present paper provides an alternative proof for the theorem in order to give insight into the structure of A as a countable union of BK spaces, that is, Banach spaces with coutinuous coordinates.

Mathematical Subject Classification

Primary: 40.46

Milestones

Received: 11 August 1967

Published: 1 October 1968