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arXiv · 2312.13920

Orthogonality of invariant measures for weighted shifts

Sophie Grivaux·Etienne Matheron·Quentin Menet
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Abstract

We introduce and study the notion of orthogonality for two operators in the context of weighted backward shifts on $\ell_p(\mathbb{Z}_+)$, $1\leq p<\infty$. 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 $\pmb u$ and $\pmb v$ implying orthogonality or non-orthogonality of the associated weighted shifts $B_{\pmb u}$ and $B_{\pmb v}$, and we investigate in some detail the case where the invariant measures are product measures.

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Sophie Grivaux, Etienne Matheron, Quentin Menet. 2023-12-21. Orthogonality of invariant measures for weighted shifts. https://arxiv.org/abs/2312.13920

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