Linear dynamics in reproducing kernel Hilbert spaces

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 $M_z^{\ast }$, 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 $M_z^{\ast }$ on tridiagonal reproducing kernel Hilbert spaces and some special classes of vector-valued analytic reproducing kernel Hilbert spaces.

23 pages. Enlarged, corrected and revised. To appear in Bulletin des Sciences Mathematiques