Subnormality and Weighted Composition Operators on L^2 Spaces
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Subnormality of bounded weighted composition operators on L2(Σ) of the form Wf= uf ◦ T, where T is a nonsingular measurable transformation on the underlying space X of a σ-finite measure space (X,Σ,μ) 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)}+∞ n=0 is a moment sequence for almost every x ∈ X, where Jn = hnEn(|u|2) ◦ T-n, hn = dμ◦T-n/dμ and En is the conditional expectation operator with respect to T-n Σ.
Abstract
Subnormality of bounded weighted composition operators on L2(Σ) of the form Wf= uf ◦ T, where T is a nonsingular measurable transformation on the underlying space X of a σ-finite measure space (X,Σ,μ) 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)}+∞ n=0 is a moment sequence for almost every x ∈ X, where Jn = hnEn(|u|2) ◦ T-n, hn = dμ◦T-n/dμ and En is the conditional expectation operator with respect to T-n Σ.
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