Hypercyclicity of weighted shifts on weighted Bergman spaces
Bibhash Kumar Das, Aneesh Mundayadan
TL;DR
This work analyzes the dynamical behavior of weighted backward shifts on weighted Bergman spaces $A^p_ ho$, focusing on hypercyclicity, chaos, and periodic vectors. By establishing sharp norm estimates for coefficient functionals $k_n$ and proving boundedness of the unweighted shift $B$, the authors derive precise criteria for hypercyclicity and chaos via the Gethner-Shapiro framework and related criteria. For radial normal weights, they obtain complete characterizations: hypercyclicity and mixing hinge on the growth of $|w_1 dots w_n|$ relative to the weight, and chaos arises under a summability condition when $1<p<2+eta$; these extend to standard and logarithmic-type weights $A^p_eta$ and $A^p_{a,b}$. The paper also extends the analysis to certain non-radial weights, providing parallel boundedness, hypercyclicity, and chaos results, thereby broadening the dynamical theory of shifts on Bergman-type spaces with general weights.
Abstract
We study the continuity, and dynamical properties (hypercyclicity, periodic vectors, and chaos) for a weighted backward shift $B_w$ on a weighted Bergman space $A^p_φ$ based on the norm estimates of coefficient functionals on $A^p_φ$. Here, the weight function $φ(z)$ is mostly radial, but our work will also involve a (non-radial) subharmonic weight. We provide a complete characterization of hypercyclic shifts $B_w$ on $A^p_φ$ when $φ$ is an (integrable) radial weight. The coefficient multipliers obtained in this paper for certain weights are new.