Univalent mappings and invariant subspaces of the Bergman and Hardy spaces

Author:
Brent J. Carswell

Journal:
Proc. Amer. Math. Soc. **131** (2003), 1233-1241

MSC (2000):
Primary 30H05, 46E20, 46E22

DOI:
https://doi.org/10.1090/S0002-9939-02-06646-7

Published electronically:
September 17, 2002

MathSciNet review:
1948115

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Abstract | References | Similar Articles | Additional Information

Abstract: In both the Bergman space and the Hardy space , the problem of determining which bounded univalent mappings of the unit disk have the wandering property is addressed. Generally, a function in has the wandering property in , where denotes either or , provided that every -invariant subspace of is generated by the orthocomplement of within . It is known that essentially every function which has the wandering property in either space is the composition of a univalent mapping with a classical inner function, and that the identity mapping has this property in both spaces. Consequently, weak-star generators of also have the wandering property in both settings. The present paper gives a partial converse to this, and shows that in both settings there is a large class of bounded univalent mappings which fail to have the wandering property.

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Additional Information

**Brent J. Carswell**

Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109

Email:
carswell@umich.edu

DOI:
https://doi.org/10.1090/S0002-9939-02-06646-7

Keywords:
Bergman space,
Hardy space,
wandering property,
invariant subspace,
multiplication operator,
reproducing kernel,
weak-star generator

Received by editor(s):
May 22, 2001

Received by editor(s) in revised form:
November 30, 2001

Published electronically:
September 17, 2002

Communicated by:
Juha M. Heinonen

Article copyright:
© Copyright 2002
American Mathematical Society