Dynamics of weighted backward shifts on certain analytic function spaces

TL;DR

This paper introduces new analytic function spaces and investigates the properties of weighted backward shift operators on them, including boundedness, similarity to compact perturbations, and dynamical behaviors like hypercyclicity and chaos.

Contribution

It defines novel Banach spaces of analytic functions and characterizes the boundedness, spectral properties, and dynamical behaviors of weighted backward shifts on these spaces.

Findings

Characterized when $B_w$ is bounded on the new spaces.

Proved $B_w$ is similar to a compact perturbation of a weighted shift.

Established conditions for hypercyclicity, mixing, and chaos of $B_w$.

Abstract

We introduce the Banach spaces $\ell^p_{a,b}$ and $c_{0,a,b}$, of analytic functions on the unit disc, having normalized Schauder bases consisting of polynomials of the form $f_n(z)=(a_n+b_nz)z^n, ~~n\geq0$, where $\{f_n\}$ is assumed to be equivalent to the standard basis in $\ell^p$ and $c_0$, respectively. We study the weighted backward shift operator $B_w$ on these spaces, and obtain necessary and sufficient conditions for $B_w$ to be bounded, and prove that, under some mild assumptions on $\{a_n\}$ and $\{b_n\}$, the operator $B_w$ is similar to a compact perturbation of a weighted backward shift on the sequence spaces $\ell^p$ or $c_0$. Further, we study the hypercyclicity, mixing, and chaos of $B_w$, and establish the existence of hypercyclic subspaces for $B_w$ by computing its essential spectrum. Similar results are obtained for a function of $B_w$ on $\ell^p_{a,b}$ and $c_{0,a,b}$.