Subnormality and Weighted Composition Operators on L2Spaces

Abstract

Subnormality of bounded weighted composition operators on L 2 () of the form Wf = uf ◦ T; where T is a nonsingular measurable transformation on the under- lying space X of a -nite measure space ( X; ; ) 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 {Jn(x)} +1 n=0 is a moment sequence for almost every x ∈ X, where Jn = hnEn(|u| 2 ) ◦T n ; hn = d ◦T n =d and En is the conditional expectation operator with respect to T n .