Localizing and Seminormal Composition Operators on L(2)
Document Type
Article
Publication Date
1994
Department
Mathematics
School
Mathematics and Natural Sciences
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.
Publication Title
Proceedings of the Royal Society of Edinburgh: Section A Mathematics
Volume
124
First Page
301
Last Page
316
Recommended Citation
Campbell, J. T.,
Hornor, W. E.
(1994). Localizing and Seminormal Composition Operators on L(2). Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 124, 301-316.
Available at: https://aquila.usm.edu/fac_pubs/7233