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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
Published electronically: September 17, 2002
MathSciNet review: 1948115
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Abstract: In both the Bergman space $A^2$ and the Hardy space $H^2$, the problem of determining which bounded univalent mappings of the unit disk have the wandering property is addressed. Generally, a function $g$ in $H^{\infty}$ has the wandering property in $X$, where $X$ denotes either $A^2$ or $H^2$, provided that every $g$-invariant subspace $M$ of $X$ is generated by the orthocomplement of $gM$ within $M$. 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 $H^{\infty}$ 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

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

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