Spectral characterization of shadowing for linear 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.