Super-Shadowing and Supercyclicity
Eric Cabezas, Manuel Saavedra
TL;DR
The paper introduces the super-shadowing property for invertible linear operators on complex Banach spaces, showing that δ-pseudotrajectories can be shadowed by rescaled orbits $\lambda_n T^n p$. It analyzes both weak and limit variants, establishing equivalences with bounded perturbation schemes and spectral decompositions, and provides a compact-operator classification: some compact operators exhibit positive (limit) super-shadowing even when positive shadowing fails. A key finding is that no surjective isometric operator on a separable Banach space with $\dim X>1$ can have the positive super-shadowing property. Extending to supercyclicity, the work develops a Birkhoff-type theorem for upper Furstenberg families, defines various notions of $\mathcal{F}$-supercyclicity, proves $\text{RSC}(T)=\text{SC}(T)$, and demonstrates residuality results for UFSC and RSC, broadening the understanding of shadowing-like properties in linear dynamics.
Abstract
We introduce the super-shadowing property in linear dynamics, where pseudotrajectories are approximated by sequences of the form $(λ_nT^nx)$, with $(λ_n)_n$ being complex scalars. For compact operators on Banach spaces, we characterize the operators that possess the positive super-shadowing property and the positive limit super-shadowing property. Additionally, we demonstrate that no surjective isometric operator on a separable Banach space $X$ with $\text{dim}(X)>1$ can exhibit the positive super-shadowing property. Finally, we provide some results on upper frequently supercyclic and reiteratively supercyclic operators.