J. Austral. Math. Soc.  72 (2002), 335-348

Strongly omnipresent operators: general conditions and applications to composition operators

L. Bernal-González   Departamento de Análisis Matemático   Facultad de Matemáticas, Apdo. 1160   Avenida Reina Mercedes   41080 Sevilla   Spain   lbernal@us.es M. C. Calderón-Moreno   Departamento de Análisis Matemático   Facultad de Matemáticas, Apdo. 1160   Avenida Reina Mercedes   41080 Sevilla   Spain   mccm@us.es and K.-G. Grosse-Erdmann   Fachbereich Mathematik   Fernuniversitat Hagen   58084 Hagen   Germany   kg.grosse-erdmann@fernuni-hagen.de

Abstract

This paper studies the concept of strongly omnipresent operators that was recently introduced by the first two authors. An operator T on the space H(G) of holomorphic functions on a complex domain G is called strongly omnipresent whenever the set of T-monsters is residual in H(G), and a T-monster is a function f such that Tf exhibits an extremely `wild' behaviour near the boundary. We obtain sufficient conditions under which an operator is strongly omnipresent, in particular, we show that every onto linear operator is strongly omnipresent. Using these criteria we completely characterize strongly omnipresent composition and multiplication operators.

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