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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles 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 1.48 .

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The commutant of analytic Toeplitz operators
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by James A. Deddens and Tin Kin Wong PDF
Trans. Amer. Math. Soc. 184 (1973), 261-273 Request permission

Abstract:

In this paper we study the commutant of an analytic Toeplitz operator. For $\phi \;\;{H^\infty }$, let $\phi = \chi F$ be its inner-outer factorization. Our main result is that if there exists $\lambda \;\epsilon \;{\text {C}}$ such that X factors as $\chi = {\chi _1}{\chi _2} \cdots {\chi _n}$, each ${\chi _i}$ an inner function, and if $F - \lambda$ is divisible by each ${\chi _i}$, then $\{ {T_\phi }\} ’ = \{ {T_\chi }\} ’ \cap \{ {T_F}\} ’$. The key step in the proof is Lemma 2, which is a curious result about nilpotent operators. One corollary of our main result is that if $\chi (z) = {z^n},n \geq 1$, then $\{ {T_\phi }\} ’ = \{ {T_\chi }\} ’ \cap \{ {T_F}\} ’$, another is that if $\phi \;\epsilon {H^\infty }$ is univalent then $\{ {T_\phi }\} ’ = \{ {T_z}\} ’$. We are also able to prove that if the inner factor of $\phi$ is $\chi (z) = {z^n},n \geq 1$, then $\{ {T_\phi }\} ’ = \{ {T_{{z^s}}}\} ’$ where s is a positive integer maximal with respect to the property that ${z^n}$ and $F(z)$ are both functions of ${z^s}$. We conclude by raising six questions.
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Additional Information
  • © Copyright 1973 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 184 (1973), 261-273
  • MSC: Primary 47B35
  • DOI: https://doi.org/10.1090/S0002-9947-1973-0324467-0
  • MathSciNet review: 0324467