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Spectra of compact composition operators


Authors: James G. Caughran and Howard J. Schwartz
Journal: Proc. Amer. Math. Soc. 51 (1975), 127-130
MSC: Primary 47B37; Secondary 30A18
DOI: https://doi.org/10.1090/S0002-9939-1975-0377579-7
MathSciNet review: 0377579
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \phi $ be holomorphic and map the open unit disk into itself, and let $ {C_\phi }:f \to f \circ \phi $ be the composition operator on $ {H^2}$ generated by $ \phi $. If $ {C_\phi }$ is a compact operator then $ (1)\phi ({z_0}) = {z_0}$ for some $ {z_0} \epsilon D$; $ (2)\sigma ({C_\phi }) = \{ \phi '{({z_0})^n}:n = 0,1,2, \ldots \} \cup \{ 0\} $.


References [Enhancements On Off] (What's this?)

  • [1] J. G. Caughran, Polynomial approximation and spectral properties of composition operators on $ {H^2}$, Indiana Univ. Math. J. 21 (1971/72), 81-84. MR 44 #4213. MR 0287006 (44:4213)
  • [2] M. A. Denjoy, Sur l'iteration des fonctions analytiques, C. R. Acad. Sci. Paris 182 (1926), 255-257.
  • [3] P. L. Duren, Theory of $ {H^p}$ spaces, Pure and Appl. Math., vol. 38, Academic Press, New York, 1970. MR 42 #3552. MR 0268655 (42:3552)
  • [4] M. Koenig, Recherches sur les intégrals de certaines équations fonctionnelles, Ann. Sci. École Norm. Sup. 3 (1884), 1-112.
  • [5] E. A. Nordgren, Composition operators, Canad. J. Math. 20 (1968), 442-449. MR 36 #6961. MR 0223914 (36:6961)
  • [6] J. V. Ryff, Subordinate $ {H^p}$ functions, Duke Math. J. 33 (1966), 347-354. MR 33 #289. MR 0192062 (33:289)
  • [7] J. H. Shapiro and P. D. Taylor, Compact, nuclear, and Hilbert-Schmidt composition operators on $ {H^2}$, Indiana Univ. Math. J. 23 (1973), 471-496. MR 0326472 (48:4816)
  • [8] J. Wolff, Sur l'iteration des fonctions..., C. R. Acad. Sci. Paris 182 (1926), 42-43, 200-201.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0377579-7
Article copyright: © Copyright 1975 American Mathematical Society

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