Frequently recurrent backward shifts

TL;DR

This paper investigates the properties of frequently recurrent backward shift operators on Fréchet sequence spaces, establishing conditions for frequent recurrence and the density of such vectors.

Contribution

It characterizes frequently recurrent backward shift operators and demonstrates the dense lineability of their frequently recurrent vectors.

Findings

If a backward shift has a non-zero frequently recurrent vector, it is frequently recurrent.

The paper provides two characterizations of frequently recurrent backward shift operators.

The set of frequently recurrent vectors is dense and lineable.

Abstract

We study frequently recurrent unilateral and bilateral backward shift operators on Fr\'echet sequence spaces. We prove that if a backward shift admits a non-zero frequently recurrent vector, then it supports a dense set of such vectors, so that the operator is frequently recurrent. As a consequence, we provide two different characterizations for frequently recurrent backward shift operators and we show dense lineability of the set of the set of frequently recurrent vectors.