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arXiv · 2511.12272

Spectral characterization of shadowing for linear operators on Hilbert spaces

Mihály Pituk
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Abstract

In this paper, we study one of the fundamental notions in dynamical systems, the shadowing of invertible (bounded and linear) operators on a Hilbert space. Although the problem of finding a spectral characterization for shadowing has been in the focus of the research for a long time, spectral criteria are available only for rather special classes of invertible operators. In this paper, we give a complete spectral characterization for the shadowing of an arbitrary invertible operator $T$ on a complex Hilbert space. It is shown that $T$ has the shadowing property if and only if its right spectrum is disjoint from the unit circle in the complex plane. As a consequence, the shadowing property for $T$ is equivalent to the uniform expansivity of its adjoint operator.

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Mihály Pituk. 2026-06-26. Spectral characterization of shadowing for linear operators on Hilbert spaces. https://arxiv.org/abs/2511.12272

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