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

Cowen, Carl C.; Kriete, Thomas L. III

J. Funct. Anal. 81, No. 2, 298-319 (1988).

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)

Keywords:

composition operator; fixed point; Möbius transformation; Embry-Lambert condition for subnormality; representation as a multiplication operator by an inner function; minimal normal extension; representation as a weighted sum of shifts

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