Subnormality and Weighted Composition Operators on L-2 Spaces

Subnormality of bounded weighted composition operators on L-2(Sigma) of the form Wf = uf o T, where T is a nonsingular measurable transformation on the underlying space X of a sigma-finite measure space (X, Sigma, mu) and u is a weight function on X; is studied. The standard moment sequence characterizations of subnormality of weighted composition operators are given. It is shown that weighted composition operators are subnormal if and only if {J(n)(x)},(+infinity)(n=0) is a moment sequence for almost every x E X, where J(n) = h(n)E(n) (vertical bar u vertical bar(2))oT(-n), h(n) = d mu oT(-n)/d mu and E-n is the conditional expectation operator with respect to T-n Sigma.