Normal extensions of subnormal composition operators. (English) Zbl 0675.47012

Lambert, Alan

Mich. Math. J. 35, No. 3, 443-450 (1988).

For a \(\sigma\)-finite measure space \((X,\Sigma,m)\) and a measurable mapping \(T\) of \(X\) onto \(X\) the corresponding composition operator \(C\) on \(L^ 2(X,\Sigma,m)\) defined by \(Cf:=f\circ T\) is considered. If \(C\) is subnormal, the minimal normal extension of \(C\) is shown to be representable as a composition operator provided that \(T\Sigma\subseteq \Sigma\) and \(m\circ T^{-1}\) as well as \(m\circ T\) are mutually absolutely continuous with respect to \(m\). This is realized by the construction of a quasi-normal composition operator which extends \(C\) even without the additional properties of \(T\).

Reviewer: Gerhard Garske (Hagen)

MSC:

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

Keywords:

quasi-normal composition operator