Holomorphic T-monsters and strongly omnipresent operators

Abstract

Assume that G is a nonempty open subset of the complex plane and that T is an operator on the linear space of holomorphic functions in G, endowed with the compact-open topology. In this paper we introduce the notions of strongly omnipresent operator and of T-monster, which are related to the wild behaviour of certain holomorphic functions near the boundary of G. T-monsters extend a concept introduced by W. Luh and K.-G. Grosse-Erdmann. After showing that T is strongly omnipresent if and only if the set of T-monsters is residual, it is proved in this paper that certain kinds of infinite order differential and antidifferential operators are strongly omnipresent, which improves some earlier nice results due to the mentioned authors.