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

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 $$T$$.

### MSC:

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

### Keywords:

quasi-normal composition operator
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