## Subnormality and composition operators on $$H^ 2$$.(English)Zbl 0669.47012

A composition operator $$C_{\phi}$$ on $$H^ 2$$ is the operator defined by $$C_{\phi}f:=f\circ \phi$$ for some nonconstant analytic function $$\phi$$ on the unit disc $${\mathbb{D}}$$ such that $$\phi$$ ($${\mathbb{D}})\subset {\mathbb{D}}$$. $$C_{\phi}$$ is normal if and only if $$\phi (z)=\alpha z$$ for some complex $$\alpha$$ such that $$| \alpha | \leq 1$$. The main result of this paper concerns conditions on $$\phi$$ implying subnormality of $$C^*_{\phi}$$. A point $$c\in {\bar {\mathbb{D}}}$$ is called fixed point of $$\phi$$ if $$\lim_{\rho \to 1^-}\phi (\rho c)=c$$. In this case $$\lim_{\rho \to 1^-}\phi '(\rho c)$$ exists and is denoted by $$\phi '(c)$$ even if $$| c| =1$$. If $$C^*_{\phi}$$ is subnormal and not normal it is shown that there exists a fixed point c such that $$| c| =1$$ and $$0<\phi '(c)<1$$. Conversely if in this case $$\phi$$ is analytic in a neighborhood of c then $$C^*_{\phi}$$ is subnormal if and only if $$\phi$$ is the Möbius transformation $$\phi (z)=[(r+s)z+(1- s)c][r(1-s)\bar cz+(1+sr)]^{-1}$$ for some r,s such hat $$0\leq r\leq 1$$, $$0<s<1$$. The analyticity request can be weakened. The proof is performed by means of the Embry-Lambert condition for subnormality. If $$C^*_{\phi}$$ is subnormal a representation as a multiplication operator by an inner function is given and for its minimal normal extension a representation as a weighted sum of shifts is obtained.
Reviewer: G.Garske

### MSC:

 47B20 Subnormal operators, hyponormal operators, etc. 47B38 Linear operators on function spaces (general) 46J15 Banach algebras of differentiable or analytic functions, $$H^p$$-spaces 30D55 $$H^p$$-classes (MSC2000)
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