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The spectra and commutants of some weighted composition operators


Author: James W. Carlson
Journal: Trans. Amer. Math. Soc. 317 (1990), 631-654
MSC: Primary 47B37; Secondary 47A05, 47A10
DOI: https://doi.org/10.1090/S0002-9947-1990-0979958-6
MathSciNet review: 979958
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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.


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DOI: https://doi.org/10.1090/S0002-9947-1990-0979958-6
Article copyright: © Copyright 1990 American Mathematical Society

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