Linear dynamics in reproducing kernel Hilbert spaces

TL;DR

This paper provides new sufficient conditions for hypercyclicity, mixing, and chaos of the adjoint of the multiplication operator in vector-valued analytic reproducing kernel Hilbert spaces, extending previous results on unilateral weighted shifts.

Contribution

It introduces new sufficient conditions for hypercyclicity, mixing, and chaos of $M_z^*$ based on kernel derivatives, and characterizes hypercyclicity in specific RKHS classes.

Findings

Established sufficient conditions for hypercyclicity, mixing, and chaos.

Provided a complete characterization of hypercyclicity in tridiagonal RKHS.

Analyzed the special case of quasi-scalar RKHS.

Abstract

Complementing earlier results on dynamics of unilateral weighted shifts, we obtain a sufficient (but not necessary, with supporting examples) condition for hypercyclicity, mixing and chaos for $M_z^*$, the adjoint of $M_z$, on vector-valued analytic reproducing kernel Hilbert spaces $\mathcal{H}$ in terms of the derivatives of kernel functions on the open unit disc $\mathbb{D}$ in $\mathbb{C}$. Here $M_z$ denotes the multiplication operator by the coordinate function $z$, that is \[ (M_z f) (w) = w f(w), \] for all $f \in \mathcal{H}$ and $w \in \mathbb{D}$. We analyze the special case of quasi-scalar reproducing kernel Hilbert spaces. We also present a complete characterization of hypercyclicity of $M_z^*$ on tridiagonal reproducing kernel Hilbert spaces and some special classes of vector-valued analytic reproducing kernel Hilbert spaces.