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Normal and quasinormal weighted composition operators

Published online by Cambridge University Press:  18 May 2009

James T. Campbell ,
Mary Embry-Wardrop ,
Richard J. Fleming  and
S. K. Narayan
James T. Campbell
Affiliation:
Department of Mathematical Sciences, Memphis State University, Memphis, TN. 38152, U.S.A.
Mary Embry-Wardrop
Affiliation:
Department of Mathematics, Central Michigan University, Mt. Pleasant, Michigan 48859, U. S. A.
Richard J. Fleming
Affiliation:
Department of Mathematics, Central Michigan University, Mt. Pleasant, Michigan 48859, U. S. A.
S. K. Narayan
Affiliation:
Department of Mathematics, Central Michigan University, Mt. Pleasant, Michigan 48859, U. S. A.
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In their paper [1], Campbell and Jamison attempted to give necessary and sufficient conditions for a weighted composition operator on an L2 space to be normal, and to be quasinormal. Those conditions, specifically Theorems I and II of that paper, are not valid (see [2] for precise comments on the other results in that paper). In this paper we present a counterexample to those theorems and state and prove characterizations of quasinormality (Theorem 1 below) and normality (Theorem 2 and Corollary 3 below). We also discuss additional examples and information concerning normal weighted composition operators which contribute to the further understanding of this class.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 33 , Issue 3 , September 1991 , pp. 275 - 279
Copyright
Copyright © Glasgow Mathematical Journal Trust 1991

References

REFERENCES

1.Campbell, J. and Jamison, J., On some classes of weighted composition operators, Glasgow Math. J. 32 (1990), 87–94.CrossRefGoogle Scholar
2.Campbell, J. and Jamison, J., Errata to: On some classes of weighted composition operators, Glasgow Math. J. 32 (1990), 261–263.CrossRefGoogle Scholar
3.Foguel, S., The ergodic theory of Markov processes, Math. Studies No. 21 (Van Nostrand Reinhold, New York, 1969).Google Scholar
4.Lambert, A., Hyponormal composition operators, Bull. London Math. Soc. 18 (1986), 395–400.CrossRefGoogle Scholar
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