The spectra and commutants of some weighted composition operators
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 by James W. Carlson PDF
 Trans. Amer. Math. Soc. 317 (1990), 631654 Request permission
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 sigmaalgebra 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 onetoone except on a finite subset of the integers.References

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Additional Information
 © Copyright 1990 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 317 (1990), 631654
 MSC: Primary 47B37; Secondary 47A05, 47A10
 DOI: https://doi.org/10.1090/S00029947199009799586
 MathSciNet review: 979958