Spectrum characterization for topological shadowing on Banach spaces

Determine whether an invertible bounded linear operator with bounded inverse on a Banach space has the topological shadowing property if and only if its spectrum is contained either in the open unit disk or in the complement of the closed unit disk.

Background

The paper studies the topological shadowing property for invertible bounded linear operators on Banach spaces, where the tolerances for pseudo-orbits and shadowing are allowed to vary continuously with the point. The authors prove the proposed spectral characterization in finite-dimensional Banach spaces, and also establish the necessary spectral condition for uniformly expansive operators and for hyperbolic operators with topological shadowing.

The unresolved problem is whether the same equivalence holds for arbitrary, possibly infinite-dimensional, Banach spaces. The conjectured spectral condition would characterize precisely when a linear operator possesses topological shadowing and would extend the finite-dimensional theorem to the general Banach-space setting.