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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The spectra and commutants of some weighted composition operators
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by James W. Carlson PDF
Trans. Amer. Math. Soc. 317 (1990), 631-654 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 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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Additional Information
  • © Copyright 1990 American Mathematical Society
  • 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