Frequently recurrent backward shifts

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.