The spectra and commutants of some weighted composition operators

Abstract:

An operator ${T_{ug}}$ on a Hilbert space $H$ of functions on a set $X$ defined by ${T_{ug}}(f) = u(f \circ g)$, where $f$ is in $H,\;u:X \to {\mathbf {C}}$ and $g:X \to X$, is called a weighted composition operator. In this paper $X$ is the set of integers and $H = {L^2}({\mathbf {Z}},\mu )$, where $\mu$ is a measure whose sigma-algebra is the power set of ${\mathbf {Z}}$. One distinguished space is ${l^2} = {L^2}({\mathbf {Z}},\mu )$, where $\mu$ is counting measure. The most important results given here are the determination of the spectrum of ${T_{ug}}$ on ${l^2}$ and a characterization of the commutant of ${T_g}$ on ${L^2}({\mathbf {Z}},\mu )$. To obtain many of the results it was necessary to assume the function $g$ to be one-to-one except on a finite subset of the integers.