## 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.

### Keywords:

composition operator; hyponormal; quasinormal