Univalent mappings and invariant subspaces of the Bergman and Hardy spaces
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- by Brent J. Carswell PDF
- Proc. Amer. Math. Soc. 131 (2003), 1233-1241 Request permission
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.References
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Additional Information
- Brent J. Carswell
- Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
- Email: carswell@umich.edu
- 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
- © Copyright 2002 American Mathematical Society
- 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
- MathSciNet review: 1948115