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

Yannic Steenbeck

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Source: arXiv

Published: Sep 4, 2026

arXiv: 2609.05735

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Source 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--López-Martínez 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 11.

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Can one feel the existence of a non-trivial invariant measure? — Mathematical Frontier Network