Univalent mappings and invariant subspaces of the Bergman and Hardy spaces
Author:
Brent J. Carswell
Journal:
Proc. Amer. Math. Soc. 131 (2003), 12331241
MSC (2000):
Primary 30H05, 46E20, 46E22
Published electronically:
September 17, 2002
MathSciNet review:
1948115
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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, weakstar 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:
http://dx.doi.org/10.1090/S0002993902066467
PII:
S 00029939(02)066467
Keywords:
Bergman space,
Hardy space,
wandering property,
invariant subspace,
multiplication operator,
reproducing kernel,
weakstar 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
