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Journal of the London Mathematical Society 2000 62(2):553-568; doi:10.1112/S0024610700001198
© 2000 by London Mathematical Society
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© The London Mathematical Society

Reducing Subspaces for a Class of Multiplication Operators

Kehe Zhu

Department of Mathematics, State University of New York Albany, NY 12222, USA, kzhu{at}math.albany.edu

Received 15 February 1999.

Let D be the open unit disk in the complex plane C. The Bergman space Formula is the Hilbert space of analytic functions f in D such that


Formula

where dA is the normalized area measure on D. If


Formula

are two functions in Formula, then the inner product of f and g is given by


Formula

We study multiplication operators on Formula induced by analytic functions. Thus for {varphi} isin H {infty}(D), the space of bounded analytic functions in D, we define


Formula

by


Formula

It is easy to check that M{phi} is a bounded linear operator on Formula with

||M{varphi}||=||{varphi}||{infty}=sup{|{varphi}(z)|:zisinD}.


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