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 Mz⁎, the adjoint of Mz, on vector-valued analytic reproducing kernel Hilbert spaces H in terms of the derivatives of kernel functions on the open unit disc D in C. Here Mz denotes the multiplication operator by the coordinate function z, that is(Mzf)(w)=wf(w), for all f∈H and w∈D. We analyze the special case of quasi-scalar reproducing kernel Hilbert spaces. We also present a complete characterization of hypercyclicity of Mz⁎ on tridiagonal reproducing kernel Hilbert spaces and some special classes of vector-valued analytic reproducing kernel Hilbert spaces.