Spectral characterization of shadowing for linear operators on Hilbert spaces

TL;DR

This work delivers a complete spectral characterization of the shadowing property for invertible operators on complex Hilbert spaces: $T$ has shadowing if and only if its right spectrum does not intersect the unit circle, a condition equivalent to the uniform expansivity of the adjoint $T^*$. The authors build on known links between hyperbolicity, expansivity, and spectral sets, and they develop a Hilbert-space–specific framework using right inverses of $\lambda I-T$ and a holomorphic/Laurent-series approach to connect spectral data to shadowing. The result unifies and sharpens prior partial criteria, providing a practical criterion in terms of $\sigma_r(T)$ and $\sigma_a(T^*)$ that can be verified for invertible operators. This advances the understanding of dynamical behavior of linear operators in infinite dimensions and clarifies the relationship between shadowing, expansivity, and spectral geometry.

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.

Theorems & Definitions (18)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Example 1.8
• Proposition 2.1
• Proposition 2.2