Orthogonality of invariant measures for weighted shifts

Sophie Grivaux; Etienne Matheron; Quentin Menet

paper

Orthogonality of invariant measures for weighted shifts

arXiv:2312.13920

Abstract

We introduce and study the notion of orthogonality for two operators in the context of weighted backward shifts on $ℓ_{p} (Z_{+})$ , $1 \leq p < \infty$ . Two continuous linear operators $T_{1}$ and $T_{2}$ acting on a Polish topological vector space $X$ are said to be orthogonal if any two Borel probability measures $m_{1}$ and $m_{2}$ on $X$ which are respectively $T_{1}$ -invariant and $T_{2}$ -invariant and satisfy $m_{1} ({0}) = m_{2} ({0}) = 0$ must be orthogonal. In this note, we provide several conditions on the weights $u$ and $v$ implying orthogonality or non-orthogonality of the associated weighted shifts $B_{u}$ and $B_{v}$ , and we investigate in some detail the case where the invariant measures are product measures.

36 p