Abstract
We consider the topological space of all composition operators, acting on certain Hilbert spaces of holomorphic functions on the unit disc, in the uniform operator topology. A sufficient condition is given for the component of a composition operator to be a singleton. A necessary condition is given for one composition operator to lie in the component of another. In addition, we prove analogous results for the component of the image of a composition operator in the Calkin algebra. Finally, we obtain some related results on the essential norm of a linear combination of composition operators.
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[B] E. Berkson, Composition operators isolated in the uniform operator topology, Proc. Amer. Math. Soc. 81 (1981) 230–232.
[H-Y] J. Hocking and G. Young, Topology, Addison-Wesley, Reading, Mass., 1961.
[L] J. Littlewood, On inequalities in the theory of functions, Proc. London Math. Soc. 23 (1925) 481–519.
[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.
[N] R. Nevanlinna, Analytic Functions, Springer-Verlag, Berlin, 1970.
[R] W. Rudin, Function Theory in the Unit Ball of Cn, Grundlehren der mathematischen Wissenschaften 241, Springer-Verlag New York, 1980.
[S1] J. Shapiro, The essential norm of a composition operator, Annals of Math. 125 (1987) 375–404.
[S2] J. Shapiro, Private communication.
[S-S] J. Shapiro and C. Sundberg, Isolation amongst the composition operators, preprint.
[S-T] J. Shapiro and P. Taylor, Compact, nuclear and Hilbert-Schmidt operators on H2, Indiana Univ. Math. J. 23 (1973) 471–496.
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Research supported in part by the National Science Foundation.
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MacCluer, B.D. Components in the space of composition operators. Integr equ oper theory 12, 725–738 (1989). https://doi.org/10.1007/BF01194560
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DOI: https://doi.org/10.1007/BF01194560