# Compact homomorphisms between algebras of analytic functions

Richard Aron; Pablo Galindo; Mikael Lindström

Studia Mathematica (1997)

- Volume: 123, Issue: 3, page 235-247
- ISSN: 0039-3223

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topAron, Richard, Galindo, Pablo, and Lindström, Mikael. "Compact homomorphisms between algebras of analytic functions." Studia Mathematica 123.3 (1997): 235-247. <http://eudml.org/doc/216391>.

@article{Aron1997,

abstract = {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.},

author = {Aron, Richard, Galindo, Pablo, Lindström, Mikael},

journal = {Studia Mathematica},

keywords = {Tsirelson space; uniform algebra of all bounded analytic functions in the open unit disk; weakly compact algebra endomorphism; composition operators; compact multiplicative operators},

language = {eng},

number = {3},

pages = {235-247},

title = {Compact homomorphisms between algebras of analytic functions},

url = {http://eudml.org/doc/216391},

volume = {123},

year = {1997},

}

TY - JOUR

AU - Aron, Richard

AU - Galindo, Pablo

AU - Lindström, Mikael

TI - Compact homomorphisms between algebras of analytic functions

JO - Studia Mathematica

PY - 1997

VL - 123

IS - 3

SP - 235

EP - 247

AB - 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.

LA - eng

KW - Tsirelson space; uniform algebra of all bounded analytic functions in the open unit disk; weakly compact algebra endomorphism; composition operators; compact multiplicative operators

UR - http://eudml.org/doc/216391

ER -

## References

top- [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. Zbl0536.46015
- [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. Zbl0717.46031
- [B1] J. Bourgain, ${H}^{\infty}$ is a Grothendieck space, Studia Math. 75 (1982), 193-226.
- [B2] J. Bourgain, New Banach space properties of the disc algebra and ${H}^{\infty}$, 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. Zbl0141.32202
- [C] S. Chae, Holomorphy and Calculus in Normed Spaces, Marcel Dekker, 1985. Zbl0571.46031
- [CM] J. Cima and A. Matheson, Completely continuous composition operators, Trans. Amer. Math. Soc. 344 (1994), 849-856. Zbl0813.47032
- [De] F. Delbaen, Weakly compact operators on the disc algebra, J. Algebra 45 (1977), 284-294. Zbl0361.46048
- [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. Zbl0484.46044
- [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. Zbl0761.46037
- [Ga] T. Gamelin, Uniform Algebras, Chelsea, 1984.
- [G] G. Garnett, Bounded Analytic Functions, Academic Press, 1981. Zbl0469.30024
- [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. Zbl0347.47032
- [H] K. Hoffman, Analytic functions and Gleason parts, Ann. of Math. 86 (1967), 74-111. Zbl0192.48302
- [K] H. Kamowitz, Compact operators of the form $u{C}_{\varphi}$, Pacific J. Math. 80 (1979), 205-211. Zbl0414.47016
- [Ma] B. MacCluer, Spectra of compact composition operators on ${H}^{p}\left({B}_{N}\right)$, 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. Zbl0826.47023
- [M1] J. Mujica, Linearization of bounded holomorphic mappings on Banach spaces, ibid. 324 (1991), 867-887. Zbl0747.46038
- [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. Zbl0243.46023
- [OW] S. Ohno and J. Wada, Compact homomorphisms on function algebras, Tokyo J. Math. 4 (1981), 105-112. Zbl0471.46035
- [R] W. Rudin, Functional Analysis, McGraw-Hill, 1991. Zbl0867.46001
- [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. Zbl0183.14102
- [Ü] A. Ülger, Some results about the spectrum of commutative Banach algebras under the weak topology and applications, Monatsh. Math. 121 (1996), 353-379. Zbl0851.46036
- [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. Zbl0744.47050

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