Frequently recurrent backward shifts

Abstract

We study frequently recurrent unilateral and bilateral backward shift operators on Fréchet 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.

v2: fixed a gap in v1. To address it, we introduce the notion of recurrent arithmetic thickening, which preserves the arithmetic structure and positive lower density. We also add a new section explaining this construction