Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-28T11:41:56.293Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Campbell, J., Embry, M.-Wardrop, Fleming, R. and Narayan, S.. Normal and quasinormal weighted composition operators. Glasgow Math. J. 33 (1991), 275–279.CrossRefGoogle Scholar
2Campbell, J. and Hornor, W.. Seminormal composition operators. J. Operator Theory (to appear).Google Scholar
3Campbell, J. and Hornor, W.. Localised conditional expectation and a subnormality criterion for the adjoint of a weighted composition operator. Houston J. Math, (to appear).Google Scholar
4Campbell, J. and Jamison, J.. On some classes of weighted composition operators. Glasgow Math. J. 32 (1990), 87–94.CrossRefGoogle Scholar
5Campbell, J. and Jamison, J.. Corrigendum to On some classes of weighted composition operators. Glasgow Math. J. 32 (1990), 261–263.CrossRefGoogle Scholar
6Conway, J.. Subnormal Operators (New York: Pitman, 1982).Google Scholar
7Embry-Wardrop, M. and Lambert, A.. Subnormality for the adjoint of a composition operator on L 2. J. Operator Theory 25 (1991), 309–318.Google Scholar
8Harrington, D. and Whitley, R.. Seminormal composition operators. J. Operator Theory 11 (1984), 125–135.Google Scholar
9Lambert, A.. Hyponormal composition operators. Bull. London Math. Soc. 18 (1986), 395–400.CrossRefGoogle Scholar
10Lambert, A.. Localising sets for sigma-algebras and related point transformations. Proc. Roy. Soc. Edinburgh Sect. A 118 (1991), 111–118.CrossRefGoogle Scholar
11Lambert, A.. Subnormal composition operators. Proc. Amer. Math. Soc. 103 (1988), 750–754.CrossRefGoogle Scholar
12Singh, R. and Kumar, A.. Characterizations of invertible, unitary, and normal composition operators. Bull. Austral. Math. Soc. 19 (1978), 81–93.CrossRefGoogle Scholar
13Whitley, R.. Normal and quasinormal composition operators. Proc. Amer. Math. Soc. 70 (1978), 114–118.CrossRefGoogle Scholar