Hyponormal composition operators. (English) Zbl 0624.47014

Lambert, Alan

Bull. Lond. Math. Soc. 18, 395-400 (1986).

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

Citations:

Zbl 0534.47017