## 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
Full Text:

### References:

 [1] [B] E. Berkson, Composition operators isolated in the uniform operator topology, Proc. Amer. Math. Soc. 81 (1981) 230-232. · Zbl 0464.30027 · doi:10.1090/S0002-9939-1981-0593463-0 [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 · doi:10.1112/plms/s2-23.1.481 [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 · doi:10.4153/CJM-1986-043-4 [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 · doi:10.2307/1971314 [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 · doi:10.1512/iumj.1973.23.23041
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.