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arXiv · 2609.05735

Can one feel the existence of a non-trivial invariant measure?

Yannic Steenbeck
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Abstract

It is shown that a bounded linear map on a complex separable Hilbert space with non-trivial invariant probability measures doesn't have to possess eigenvalues. This resolves a question indicated by Flytzanis in 1995 and concretely asked by Grivaux--L\'opez-Mart\'inez from 2023 resp. Grivaux--Matheron--Menet from 2021. Still, as a positive result, we prove that every bounded linear operator on a separable complex Banach space for which a non-fixing invariant probability measure exists, has to have some approximate point spectrum on the unit circle minus $1$.

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Yannic Steenbeck. 2026-09-04. Can one feel the existence of a non-trivial invariant measure?. https://arxiv.org/abs/2609.05735

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