Super-Shadowing and Supercyclicity

Abstract

We introduce the super-shadowing property in linear dynamics, where pseudotrajectories are approximated by sequences of the form $(\lambda_nT^nx)$, with $(\lambda_n)_n$ 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.