Journal of the Australian Mathematical Society

Research Article

U-Operators

L. Bernal-Gonzáleza1 and J. A. Prado-Tenderoa1

a1 Departmento de Análisis Matemático, Facultad de Mathemáticas, Apdo. 1160, Avenida Reina Mercedes, 41080 Sevilla, Spain e-mail: lbernal@us.es, tendero@us.es

Abstract

Inspired by a statement of W. Luh asserting the existence of entire functions having together with all their derivatives and antiderivatives some kind of additive universality or multiplicative universality on certain compact subsets of the complex plane or of, respectively, the punctured complex plane, we introduce in this paper the new concept of U-operators, which are defined on the space of entire functions. Concrete examples, including differential and antidifferential operators, composition, multiplication and shift operators, are studied. A result due to Luh, Martirosian and Müller about the existence of universal entire functions with gap power series is also strengthened.

(Received February 05 2003)

(Revised September 16 2003)

2000 Mathematics subject classification

  • primary 30E10;
  • secondary 47A16;
  • 47B33;
  • 47B38;
  • 47E05;
  • 47G10

Keywords and phrases

  • universal function;
  • gap series;
  • composition operator;
  • differential operator;
  • integral operator;
  • Taylor shift;
  • U-operator