## Hyponormal composition operators.(English)Zbl 0624.47014

Let (X,$$\Sigma$$,m) be a complete $$\sigma$$-finite measure space, and let T be a $$\Sigma$$-measurable mapping in X such that $$m\circ T^{-1}$$ is absolutely continuous with respect to m. The corresponding weighted composition operator W on $$L^ 2(X,\Sigma,m)$$ generated by the weight function $$\phi$$ is defined by $$Wf:=\phi f\circ T$$. A measure theoretic characterization of hyponormality for such operators is given. This generalizes a result of D. Harrington and R. Whitley [J. Oper. Theory 11, 125-135 (1984; Zbl 0534.47017)], who considered the case of unweighted composition operators (i.e. $$\phi =1)$$. An example illuminating the connection to isometries is given.
Reviewer: G.Garske

### MSC:

 47B20 Subnormal operators, hyponormal operators, etc. 47B38 Linear operators on function spaces (general)

### Keywords:

weighted composition operator; hyponormality

Zbl 0534.47017
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