J Austral Math Soc Ser A 45 pp195--203, 1988.

Bounded Spherical Functions for Abstract Operators

Joel M. Cohen

Abstract

The aim of this paper is to study the spherical functions associated to an operator. These functions can be thought of abstractly as being eigenfunctions of the operator which can be expressed in terms of the operator. The meaning of these properties will be made precise as will a notion of boundedness. The results are obtained by studying a specific shift operator on the algebra of functionals on the complex polynomial ring. For the class studied, we obtain ellipses of eigenvalues for which there exist bounded spherical functions. As an application of the results, we study radial functions on discrete groups.

1980 AMS Subject Classification: primary 43A90; secondary 42C10

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Authors

Joel M. Cohen

Department of Mathematics, University of Bari, 70125 Bari, Italy.

and

Department of Mathematics, University of Maryland, College Park, Maryland 20742, U.S.A.