Operators with dense, invariant, cyclic vector manifolds

• Published in: Journal of functional analysis Vol. 98; no. 2; pp. 229 - 269
• Main Authors: Godefroy, Gilles, Shapiro, Joel H
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.06.1991
• ISSN: 0022-1236; 1096-0783
• DOI: 10.1016/0022-1236(91)90078-J

We study a class of Banach space operators patterned after the weighted backward shifts on Hilbert space, and show that any non-scalar operator in the commutant of one of these “generalized backward shifts” has a dense, invariant linear manifold whose non-zero members are cyclic vectors. Under appropriate extra hypotheses on the commuting operator, stronger forms of cyclicity are possible, the most extreme being hypercyclicity (density of an orbit). Motivated by these results, we examine the cyclic behavior of two seemingly unrelated classes of operators: adjoint multiplications on Hilbert spaces of holomorphic functions, and differential operators on the Fréchet space of entire functions. We show that each of these operators (other than the scalar multiples of the identity) possesses a dense, invariant linear submanifold each of whose non-zero elements is hypercyclic. Finally, we explore some connections with dynamics; many of the hypercyclic operators discussed here are, in at least one of the commonly accepted senses of the word, “chaotic.”