Nuclear Fréchet Spaces of Entire Functions with Transitive Differentiation

Abstract

A continuous . linear operator on an infinite dimensional complex topological vector space is called transitive, if it has no closed invariant subspaces which are different from the zero space and the whole space. The existence of transitive operators on certain nonreflexive Banach spaces was established by Enflo [6] and Read [14J. On the other hand, the existence of such operators on a reflexive Banach space, in particular on a Hilbert space, is still an open problem. As shown in [2], the problem for Hilbert spaces is equivalent to the problem of whether there exists a Hilbert space of entire functions with reproducing kernel, on which the differentiation operator is continuous and transitive. In this paper we consider the transitivity problem of the differentiation operator on certain nuclear Frechet spaces of entire functions. Our main result is that, for 0 < a < 1, the vector space Fa of all entire functions f for which the norms