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: For a function *f* in of the unit disk, the operator on of multiplication by *f* will be denoted by and its commutant by . For a finite Blaschke product *B*, a representation of an operator in as a function on the Riemann surface of motivates work on more general functions. A theorem is proved which gives conditions on a family of functions which imply that there is a function *h* such that . As a special case of this theorem, we find that if the inner factor of is a finite Blaschke product for some *c* in the disk, then there is a finite Blaschke product *B* with . Necessary and sufficient conditions are given for an operator to commute with when *f* is a covering map (in the sense of Riemann surfaces). If *f* and *g* are in and , then . This paper introduces a class of functions, the -ancestral functions, for which the converse is true. If *f* and *g* are -ancestral functions, then unless where *h* is univalent. It is shown that inner functions and covering maps are -ancestral functions, although these do not exhaust the class. Two theorems are proved, each giving conditions on a function *f* which imply that does not commute with nonzero compact operators. It follows from one of these results that if *f* is an -ancestral function, then does not commute with any nonzero compact operators.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1978-0482347-9

Keywords:
Toeplitz operator,
commutant,
,
analytic function,
inner function,
universal covering map

Article copyright:
© Copyright 1978
American Mathematical Society