Vol. 27, No. 1, 1968

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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

Let S be the space of all complex sequences. An element u = {un}n=0 of S is called analytic if for some constant M > 0, |un|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 yn = m=0cnmum maps A into A if and only if for every p > 0 there exists a q > 0 and a constant M > 0 such that |cnm|Mpm∕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=0cnmzmyn 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
Received: 11 August 1967
Published: 1 October 1968
Louise Arakelian Raphael