# Subnormality and Weighted Composition Operators on L2 Spaces

• Accepted : 2013.10.23
• Published : 2015.06.23
• 19 3

#### Abstract

Subnormality of bounded weighted composition operators on $L^2({\Sigma})$ of the form $Wf=uf{\circ}T$, 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 $\{J_n(x)\}^{+{\infty}}_{n=0}$ is a moment sequence for almost every $x{{\in}}X$, where $J_n=h_nE_n({\mid}u{\mid}^2){\circ}T^{-n}$, $h_n=d{\mu}{\circ}T^{-n}/d{\mu}$ and $E_n$ is the conditional expectation operator with respect to $T^{-n}{\Sigma}$.

#### Keywords

Subnormal;Weighted composition operators;Conditional expectation;Moment sequence

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#### Cited by

1. Quasinormal extensions of subnormal operator-weighted composition operators in ℓ 2 -spaces vol.452, pp.1, 2017, https://doi.org/10.1016/j.jmaa.2017.02.057