Dynamics of weighted shifts on $\ell^p$-sums and $c_0$-sums

TL;DR

This paper studies generalized weighted shift operators on Banach space sums, analyzing their dynamical behaviors like hypercyclicity and chaos, and compares these properties with the individual operators involved.

Contribution

It introduces a framework for weighted shifts with operator weights on Banach space sums and explores their complex dynamical properties, extending classical linear dynamics criteria.

Findings

Characterization of hypercyclicity and chaos for generalized shifts

Comparison of dynamical properties between individual operators and shifts

Interpretation of classical criteria in the context of these generalized shifts

Abstract

We investigate a generalization of weighted shifts where each weight $w_k$ is replaced by an operator $T_k$ going from a Banach space $X_k$ to another one $X_{k-1}$. We then look if the obtained shift operator $B_{(T_k)}$ defined on the $\ell^p$-sum (or the $c_0$-sum) of the spaces $X_k$ is hypercyclic, weakly mixing, mixing, chaotic or frequently hypercyclic. We also compare the dynamical properties of $T$ and of the corresponding shift operator $B_T$. Finally, we interpret some classical criteria in Linear Dynamics in terms of the dynamical properties of a shift operator.