Recurrent Linear Operators

TL;DR

This paper investigates recurrence properties of linear operators on Banach spaces, providing characterizations for various classes and reducing the problem to unitary operators in Hilbert spaces, while also exploring product recurrence and open questions.

Contribution

It offers new characterizations of recurrence for classes like weighted shifts, composition, and multiplication operators, and links recurrence to unitary operators in Hilbert spaces.

Findings

Recurrence characterized for weighted shifts, composition, and multiplication operators.

On separable complex Hilbert spaces, recurrence reduces to unitary operators.

Open questions posed on product recurrence.

Abstract

We study the notion of recurrence and some of its variations for linear operators acting on Banach spaces. We characterize recurrence for several classes of linear operators such as weighted shifts, composition operators and multiplication operators on classical Banach spaces. We show that on separable complex Hilbert spaces the study of recurrent operators reduces, in many cases, to the study of unitary operators. Finally, we study the notion of product recurrence and state some relevant open questions.