The commutant of an analytic Toeplitz operator

Author:
Carl C. Cowen

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

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Abstract | References | Similar Articles | Additional Information

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.

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Additional Information

Keywords:
Toeplitz operator,
commutant,
<!– MATH ${H^\infty },{H^2}$ –> <IMG WIDTH="77" HEIGHT="43" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="${H^\infty },{H^2}$">,
analytic function,
inner function,
universal covering map

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© Copyright 1978
American Mathematical Society