Vol. 30, No. 2, 1969

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A subcollection of algebras in a collection of Banach spaces

Robert Paul Kopp

Vol. 30 (1969), No. 2, 433–435
Abstract

Let D(p,r) with 1 p < and −∞ < r < +denote the Banach space consisting of certain analytic functions f(z) defined in the unit disk. A function f(z) = n=0anzn is a member of D(p,r) if and only if

∑∞       r   p
(n +1) |an |< ∞.
n=0

We define the norm of f in D(p,r) by

        ∑∞       r   p
∥f∥p.r = (  (n+ 1) |an| )1∕p.
n=0

By the product of two functions f and g in D(p,r) we shall mean their product as functions, i.e., [f.g](z) = f(z)g(z). The purpose of this paper is to discover which of the spaces D(p,r) are algebras.

Mathematical Subject Classification
Primary: 46.55
Milestones
Received: 26 February 1968
Published: 1 August 1969
Authors
Robert Paul Kopp