Normal and quasinormal weighted composition operators

James T. Campbell, Mary Embry-Wardrop, Richard J. Fleming, S. K. Narayan
1991 Glasgow Mathematical Journal  
Introduction. In their paper [1], Campbell and Jamison attempted to give necessary and sufficient conditions for a weighted composition operator on an L 2 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
more » ... 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. In what follows, (X,1.,ii) will be a complete a-finite measure space. T:X-*X will be a measurable transformation of X onto itself with the properties that the measure fi°T~l is absolutely continuous with respect to fi, and (i°T~l is finite. We set h = d(i°T~lldfi. By T~XY we mean the relative completion of the a-algebra generated by {T~lA:A e l } . With the space X and the measure /x fixed, if F c 2 is a a-algebra we write L 2 (F) as the usual equivalence classes of F measurable functions whose modulus squared is integrable over X. We denote by E:L 2 (Z)-> L 2 (T~11,) the so-called conditional expectation operator with respect to the a-algebra T~l2. More generally, £ ( / ) may be defined for bounded measurable functions / or non-negative measurable functions / ; for details on the properties of E see [1], [3], [4]. Given a 2-measurable function :X^> is defined by Glasgow Math. J. 33 (1991) 275-279. https://doi.
doi:10.1017/s0017089500008338 fatcat:qzmyon4uqndntjx34ctlrqioxq