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Seminormal composition operators. (English) Zbl 0534.47017

A composition operator \(C_ T\) on \(L^ 2(S,\Sigma,\mu)\) is a bounded linear operator induced by a map \(T:S\to S\) via \(C_ Tf(s)=f(Ts).\) The adjoint \(C^*_ T\) is hyponormal iff
a) all measurable sets intersected with the support of \(h=d\mu T^{- 1}/d\mu\) are (essentially) in \(T^{-1}(\Sigma)\) and
b) \(h{\mathbb{O}}T\geq h\) a.e.
The adjoint \(C^*_ T\) is quasinormal iff condition a) holds and \(h{\mathbb{O}}T=h\) a.e. on the support of h;
Corollary: \(C_ T\) is normal iff \(C^*_ T\) is quasinormal and \(h>0\) a.e.
The condition c) \(h\geq h{\mathbb{O}}T\) implies that \(C_ T\) is hyponormal. If \(C_ T\) is hyponormal and d) h is \(T^{-1}(\Sigma)\) measurable, then c) holds. That c) and d) both hold is equivalent to a norm condition similar to, but stronger than, one characterizing hyponormal \(C_ T\). A condition weaker than c), but still implying hyponormality, is given by the conditional expectation inequality \[ \int_{A}(h{\mathbb{O}}T- (h{\mathbb{O}}T)^ 2/h)d\mu \geq 0\quad for\quad all\quad A\quad in\quad T^{-1}(\Sigma). \]

MSC:

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