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Components in the space of composition operators. (English) Zbl 0685.47027

Let H be a Hilbert space of holomorphic functions on the unit disc \({\mathbb{D}}\) of the complex plane. For \(\alpha >1\), \(D_{\alpha}\) denotes the set \[ \{f\quad holomorphic\quad on\quad {\mathbb{D}}:\quad \| f\|^ 2_{\alpha}=\frac{\alpha -1}{\pi}\int_{{\mathbb{D}}}| f(z)|^ 2(1-| z|^ 2)^{\alpha -2\quad} dA(z)<\infty \}, \] and \(D_ 1\) is identified as \(H^ 2({\mathbb{D}})\). If \(\phi\) : \({\mathbb{D}}\to {\mathbb{D}}\) is holomorphic, then the composition operator \(C_{\phi}\) is defined by \(C_{\phi}(f)=f\circ \phi\). In the main results of this paper, the author states (i) (Corollary 2.3) that if \(\phi\) has finite angular derivatives on a set of positive measure, \(\psi\neq \phi\), then for some number s, \(\| C_{\psi}- C_{\phi}\|_ e>s^{-\alpha}\), so that the component of \(C_{\phi}\) is the singleton \(\{C_{\phi}\}\), and (ii) (Theorem 2.4) that if \(C_{\psi}\) is in the component containing \(C_{\phi}\), then \(\psi\) and \(\phi\) have the same data at any point \(e^{i\theta}\) such that \(| \phi '(e^{i\theta})| <\infty\).
Reviewer: G.O.Okikiolu

MSC:

47B38 Linear operators on function spaces (general)
47L05 Linear spaces of operators
46E20 Hilbert spaces of continuous, differentiable or analytic functions
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