Universal vectors for operators on spaces of holomorphic functions

Abstract

A vector x x in a linear topological space X X is called universal for a linear operator T T on X X if the orbit { T n x : n ≥ 0 } \{ {T^n}x:n \geq 0\} is dense in X X . Our main result gives conditions on T T and X X which guarantee that T T will have universal vectors. It applies to the operators of differentiation and translation on the space of entire functions, where it makes contact with Pólya’s theory of final sets; and also to backward shifts and related operators on various Hilbert and Banach spaces.