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Localising and seminormal composition operators on L2

Published online by Cambridge University Press:  14 November 2011

James T. Campbell  and
William E. Hornor
James T. Campbell
Affiliation:
Department of Mathematical Sciences, Memphis State University, Memphis, TN 38152, U.S.A.
William E. Hornor
Affiliation:
Department of Mathematics, The University of Southern Mississippi, P.O. Box 5045, Hattiesburg, MS 39406-5045, U.S.A.
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Abstract

Let (X, ∑, μ) denote a σ-finite measure space. We show that the kernel condition on a weighted composition operator acting on L2(X, ∑, μ), which is necessary for hyponormality of the adjoint, implies that a certain subset of X has the localising property defined by Lambert. For operators satisfying this condition, we find a reducing subspace whose orthocomplement in L2 is annihilated by both the operator and its adjoint, allowing us to obtain characterisations of seminormality for the operator by looking only at the restriction to the reducing subspace. This simplifies the analysis significantly, giving transparent characterisations for the hyponormality and quasinormality of the adjoint, as well as a characterisation of normality for the operator which does not require the computation of any conditional expectations. Several examples are given. We then characterise the semi-hyponormal class for both the operator and its adjoint.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 2 , 1994 , pp. 301 - 316
Copyright
Copyright © Royal Society of Edinburgh 1994

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