Compact homomorphisms between algebras of analytic functions

• Autorzy: Richard Aron , Pablo Galindo , Mikael Lindström
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 46E15: Banach spaces of continuous, differentiable or analytic functions
• 46G20: Infinite-dimensional holomorphy
• 46J15: Banach algebras of differentiable or analytic functions, H p -spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1997
• Tom: 123
• Numer: 3
• Strony: 235-247

Daty

wydano

1997

otrzymano

1996-02-08

poprawiono

1996-07-30

Twórcy

autor

Richard Aron

• Department of Mathematics, Kent State University, Kent, Ohio 44242, U.S.A.

autor

Pablo Galindo

• Departamento de Análisis Matemático, Universidad de Valencia, 46100 Burjasot, Valencia, Spain

autor

Mikael Lindström

• Department of Mathematics, Åbo Akademi University, FIN-20500 Åbo, Finland

Bibliografia

• [AAD] R. Alencar, R. M. Aron and S. Dineen, A reflexive space of holomorphic functions in infinitely many variables, Proc. Amer. Math. Soc. 90 (1984), 407-411.
• [ACG] R. M. Aron, B. J. Cole and T. W. Gamelin, Spectra of algebras of analytic functions on a Banach space, J. Reine Angew. Math. 415 (1991), 51-93.
• [B1] J. Bourgain, $H^∞$ is a Grothendieck space, Studia Math. 75 (1982), 193-226.
• [B2] J. Bourgain, New Banach space properties of the disc algebra and $H^∞$, Acta Math. 152 (1984), 1-48.
• [BP] A. Brown and C. Pearcy, Spectra of tensor products of operators, Proc. Amer. Math. Soc. 17 (1966), 162-166.
• [C] S. Chae, Holomorphy and Calculus in Normed Spaces, Marcel Dekker, 1985.
• [CM] J. Cima and A. Matheson, Completely continuous composition operators, Trans. Amer. Math. Soc. 344 (1994), 849-856.
• [De] F. Delbaen, Weakly compact operators on the disc algebra, J. Algebra 45 (1977), 284-294.
• [Di] J. Diestel, Sequences and Series in Banach Spaces, Grad. Texts in Math. 92, Springer, 1984.
• [D] S. Dineen, Complex Analysis in Locally Convex Spaces, North-Holland, 1981.
• [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.
• [Ga] T. Gamelin, Uniform Algebras, Chelsea, 1984.
• [G] G. Garnett, Bounded Analytic Functions, Academic Press, 1981.
• [Go] H. Goldmann, Uniform Fréchet Algebras, North-Holland, 1990.
• [HS] T. Hayden and T. Suffridge, Fixed points of holomorphic maps in Banach spaces, Proc. Amer. Math. Soc. 60 (1976), 95-105.
• [H] K. Hoffman, Analytic functions and Gleason parts, Ann. of Math. 86 (1967), 74-111.
• [K] H. Kamowitz, Compact operators of the form $uC_ϕ$, Pacific J. Math. 80 (1979), 205-211.
• [Ma] B. MacCluer, Spectra of compact composition operators on $H^p(B_N)$, Analysis 4 (1984), 87-103.
• [MM] K. Madigan and A. Matheson, Compact composition operators on the Bloch space, Trans. Amer. Math. Soc. 347 (1995), 2679-2687.
• [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.
• [S] M. Schechter, On the spectra of operators on tensor products, J. Funct. Anal. 4 (1969), 95-99.
• [Ü] A. Ülger, Some results about the spectrum of commutative Banach algebras under the weak topology and applications, Monatsh. Math. 121 (1996), 353-379.
• [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.