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THE JOINT SIMILARITY PROBLEM FOR WEIGHTED BERGMAN SHIFTS

Published online by Cambridge University Press:  05 February 2002

Sarah H. Ferguson  and
Srdjan Petrovic
Sarah H. Ferguson
Affiliation:
Department of Mathematics, Wayne State University, Detroit, MI 48202, USA ([email protected])
Srdjan Petrovic
Affiliation:
Department of Mathematics and Statistics, Western Michigan University, Kalamazoo, MI 49008, USA ([email protected])
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Abstract

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We solve a joint similarity problem for pairs of operators of Foias–Williams/Peller type on weighted Bergman spaces. We show that for the single operator, the Hardy space theory established by Bourgain and Aleksandrov–Peller carries over to weighted Bergman spaces, by establishing the relevant weak factorizations. We then use this fact, together with a recent dilation result due to the first author and Rochberg, to show that a commuting pair of such operators is jointly polynomially bounded if and only if it is jointly completely polynomially bounded. In this case, the pair is jointly similar to a pair of contractions by Paulsen’s similarity theorem.

AMS 2000 Mathematics subject classification: Primary 47B35; 47B47

Keywords

Bergman spacesHankel operatorscompletely bounded maps
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 45 , Issue 1 , February 2002 , pp. 117 - 139
Copyright
Copyright © Edinburgh Mathematical Society 2002