## 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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### References:

 [1] [B] E. Berkson, Composition operators isolated in the uniform operator topology, Proc. Amer. Math. Soc. 81 (1981) 230-232. · Zbl 0464.30027 [2] [H-Y] J. Hocking and G. Young, Topology, Addison-Wesley, Reading, Mass., 1961. [3] [L] J. Littlewood, On inequalities in the theory of functions, Proc. London Math. Soc. 23 (1925) 481-519. · JFM 51.0247.03 [4] [M-S] B. MacCluer and J. Shapiro, Angular derivatives and compact composition operators on Hardy and Bergman spaces, Canadian J. Math. 38 (1986) 878-906. · Zbl 0608.30050 [5] [N] R. Nevanlinna, Analytic Functions, Springer-Verlag, Berlin, 1970. · Zbl 0199.12501 [6] [R] W. Rudin, Function Theory in the Unit Ball of Cn, Grundlehren der mathematischen Wissenschaften 241, Springer-Verlag New York, 1980. [7] [S1] J. Shapiro, The essential norm of a composition operator, Annals of Math. 125 (1987) 375-404. · Zbl 0642.47027 [8] [S2] J. Shapiro, Private communication. [9] [S-S] J. Shapiro and C. Sundberg, Isolation amongst the composition operators, preprint. · Zbl 0732.30027 [10] [S-T] J. Shapiro and P. Taylor, Compact, nuclear and Hilbert-Schmidt operators on H2, Indiana Univ. Math. J. 23 (1973) 471-496. · Zbl 0276.47037
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