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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 commutant of an analytic Toeplitz operator
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by Carl C. Cowen PDF
Trans. Amer. Math. Soc. 239 (1978), 1-31 Request permission

Abstract:

For a function f in ${H^\infty }$ of the unit disk, the operator on ${H^2}$ of multiplication by f will be denoted by ${T_f}$ and its commutant by $\{ {T_f}\} ’$. For a finite Blaschke product B, a representation of an operator in ${\{ {T_B}\}’}$ as a function on the Riemann surface of ${B^{ - 1}} \circ B$ motivates work on more general functions. A theorem is proved which gives conditions on a family $\mathcal {F}$ of ${H^\infty }$ functions which imply that there is a function h such that $\{ {T_h}\} ’ = { \cap _{f \in \mathcal {F}}}\{ {T_f}\} ’$. As a special case of this theorem, we find that if the inner factor of $f - f(c)$ is a finite Blaschke product for some c in the disk, then there is a finite Blaschke product B with $\{ {T_f}\} ’ = \{ {T_B}\} ’$. Necessary and sufficient conditions are given for an operator to commute with ${T_f}$ when f is a covering map (in the sense of Riemann surfaces). If f and g are in ${H^\infty }$ and $f = h \circ g$, then $\{ {T_f}\} ’ \supset \{ {T_g}\} ’$. This paper introduces a class of functions, the ${H^2}$-ancestral functions, for which the converse is true. If f and g are ${H^2}$-ancestral functions, then $\{ {T_f}\} ’ \ne \{ {T_g}\} ’$ unless $f = h \circ g$ where h is univalent. It is shown that inner functions and covering maps are ${H^2}$-ancestral functions, although these do not exhaust the class. Two theorems are proved, each giving conditions on a function f which imply that ${T_f}$ does not commute with nonzero compact operators. It follows from one of these results that if f is an ${H^2}$-ancestral function, then ${T_f}$ does not commute with any nonzero compact operators.
References
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Additional Information
  • © Copyright 1978 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 239 (1978), 1-31
  • MSC: Primary 47B35; Secondary 30A78
  • DOI: https://doi.org/10.1090/S0002-9947-1978-0482347-9
  • MathSciNet review: 0482347