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1997 | 123 | 3 | 235-247
Tytuł artykułu

Compact homomorphisms between algebras of analytic functions

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We prove that every weakly compact multiplicative linear continuous map from $H^∞(D)$ into $H^∞(D)$ is compact. We also give an example which shows that this is not generally true for uniform algebras. Finally, we characterize the spectra of compact composition operators acting on the uniform algebra $H^∞(B_E)$, where $B_E$ is the open unit ball of an infinite-dimensional Banach space E.
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  • Department of Mathematics, Kent State University, Kent, Ohio 44242, U.S.A., aron@mcs.kent.edu
  • Departamento de Análisis Matemático, Universidad de Valencia, 46100 Burjasot, Valencia, Spain, galindo@uv.es
  • Department of Mathematics, Åbo Akademi University, FIN-20500 Åbo, Finland, mlindstr@ra.abo.fi
Bibliografia
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  • [B2] J. Bourgain, New Banach space properties of the disc algebra and $H^∞$, Acta Math. 152 (1984), 1-48.
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  • [C] S. Chae, Holomorphy and Calculus in Normed Spaces, Marcel Dekker, 1985.
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  • [EH] C. Earle and R. Hamilton, A fixed point theorem for holomorphic mappings, in: Global Analysis, Proc. Sympos. Pure Math. 16, Amer. Math. Soc., 1970, 61-65.
  • [GRW] J. E. Galé, T. J. Ransford and M. C. White, Weakly compact homomorphisms, Trans. Amer. Math. Soc. 331 (1992), 815-824.
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  • [M1] J. Mujica, Linearization of bounded holomorphic mappings on Banach spaces, ibid. 324 (1991), 867-887.
  • [M2] J. Mujica, Complex Analysis in Banach Spaces, North-Holland, 1986.
  • [N] K. Ng, On a theorem of Dixmier, Math. Scand. 29 (1971), 279-280.
  • [OW] S. Ohno and J. Wada, Compact homomorphisms on function algebras, Tokyo J. Math. 4 (1981), 105-112.
  • [R] W. Rudin, Functional Analysis, McGraw-Hill, 1991.
  • [Sa] D. Sarason, Weak Compactness of Holomorphic Composition Operators on $H^1$, Lecture Notes in Math. 1511, Springer, Berlin, 1990.
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  • [W] K. Włodarczyk, On the existence and uniqueness of fixed points for holomorphic maps in complex Banach spaces, Proc. Amer. Math. Soc. 112 (1991), 983-987.
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Bibliografia
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bwmeta1.element.bwnjournal-article-smv123i3p235bwm
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