Spectral characterization of shadowing for linear operators on Hilbert spaces

TL;DR

This paper provides a complete spectral characterization of shadowing for invertible operators on Hilbert spaces, showing it occurs iff the operator's right spectrum is disjoint from the unit circle, linking shadowing to spectral properties.

Contribution

It offers the first comprehensive spectral criterion for shadowing in arbitrary invertible operators on Hilbert spaces, extending previous special-case results.

Findings

Shadowing occurs iff the right spectrum is disjoint from the unit circle.

Shadowing is equivalent to the uniform expansivity of the adjoint operator.

Provides a spectral criterion applicable to all invertible operators on Hilbert spaces.

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.