Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 30H05, 46E20, 46E22

Retrieve articles in all journals with MSC (2000): 30H05, 46E20, 46E22


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/S0002-9939-02-06646-7
PII: S 0002-9939(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