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1995 | 114 | 2 | 159-180
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Banach space properties of strongly tight uniform algebras

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We use the work of J. Bourgain to show that some uniform algebras of analytic functions have certain Banach space properties. If X is a Banach space, we say X is strong} if X and X* have the Dunford-Pettis property, X has the Pełczyński property, and X* is weakly sequentially complete. Bourgain has shown that the ball-algebras and the polydisk-algebras are strong Banach spaces. Using Bourgain's methods, Cima and Timoney have shown that if K is a compact planar set and A is R(K) or A(K), then A and A* have the Dunford-Pettis property. Prior to the work of Bourgain, it was shown independently by Wojtaszczyk and Delbaen that R(K) and A(K) have the Pełczyński property for special classes of sets K. We show that if A is R(K) or A(K), where K is arbitrary, or if A is A(D) where D is a strictly pseudoconvex domain with smooth $C^2$ boundary in $ℂ^n$, then A is a strong Banach space. More generally, if A is a uniform algebra on a compact space K, we say A is strongly tight} if the Hankel-type operator $S_g: A → C/A$ defined by f → fg + A is compact for every g ∈ C(K). Cole and Gamelin have shown that R(K) and A(K) are strongly tight when K is arbitrary, and their ideas can be used to show A(D) is strongly tight for the domains D considered above. We show strongly tight uniform algebras are strong Banach spaces.
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Bibliografia
  • [1] C. Bessaga and A. Pełczyński, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151-164.
  • [2] J. Bourgain, On weak completeness of the dual of spaces of analytic and smooth functions, Bull. Soc. Math. Belg. Sér. B 35 (1) (1983), 111-118.
  • [3] J. Bourgain, The Dunford-Pettis property for the ball-algebras, the polydisc algebras and the Sobolev spaces, Studia Math. 77 (3) (1984), 245-253.
  • [4] J. Bourgain, New Banach space properties of the disc algebra and $H^∞$, Acta Math. 152 (1984), 1-48.
  • [5] J. Bourgain and F. Delbaen, A class of special $ℒ_∞$ spaces, ibid. 145 (1980), 155-176.
  • [6] J. Chaumat, Une généralisation d'un théorème de Dunford-Pettis, Université de Paris XI, U.E.R. Mathématique, preprint no. 85, 1974.
  • [7] J. A. Cima and R. M. Timoney, The Dunford-Pettis property for certain planar uniform algebras, Michigan Math. J. 34 (1987), 99-104.
  • [8] B. J. Cole and T. W. Gamelin, Tight uniform algebras and algebras of analytic functions, J. Funct. Anal. 46 (1982), 158-220.
  • [9] B. J. Cole and R. M. Range, A-measures on complex manifolds and some applications, ibid. 11 (1972), 393-400.
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  • [12] F. Delbaen, Weakly compact operators on the disc algebra, J. Algebra 45 (1977), 284-294.
  • [13] F. Delbaen, The Pełczyński property for some uniform algebras, Studia Math. 64 (1979), 117-125.
  • [14] J. Diestel, A survey of results related to the Dunford-Pettis property, in: Proc. Conf. on Integration, Topology, and Geometry in Linear Spaces, W. H. Graves (ed.), Amer. Math. Soc., Providence, R.I., 1980, 15-60.
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  • [17] T. W. Gamelin, Uniform algebras on plane sets, in: Approximation Theory, Academic Press, New York, 1973, 100-149.
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  • [19] A. Grothendieck, Sur les applications linéaires faiblement compactes d'espaces du type C(K), Canad. J. Math. 5 (1953), 129-173.
  • [20] G. M. Henkin, The Banach spaces of analytic functions in a sphere and in a bicylinder are not isomorphic, Funktsional. Anal. i Prilozhen. 2 (4) (1968), 82-91 (in Russian); English transl.: Functional Anal. Appl. 2 (4) (1968), 334-341.
  • [21] G. M. Henkin, Integral representations of functions holomorphic in strictly pseudoconvex domains and some applications, Mat. Sb. 78 (120) (4) (1969), 611-632 (in Russian); English transl.: Math. USSR-Sb. 7 (1969), 597-616.
  • [22] N. Kerzman, Hölder and $L^p$ estimates for solutions of $∂̅u = f$ in strongly pseudoconvex domains, Comm. Pure Appl. Math. 24 (1971), 301-379.
  • [23] S. V. Kisliakov, On the conditions of Dunford-Pettis, Pełczyński, and Grothendieck, Dokl. Akad. Nauk SSSR 225 (1975), 1252-1255 (in Russian); English transl.: Soviet Math. Dokl. 16 (1975), 1616-1621.
  • [24] S. Li and B. Russo, The Dunford-Pettis property for some function algebras in several complex variables, preprint, Univ. of California at Irvine, 1992.
  • [25] J. D. McNeal, On sharp Hölder estimates for solutions of the ∂̅-equations, in: Proc. Sympos. Pure Math. 52, Part 3, Amer. Math. Soc., Providence, R.I., 1991, 277-285.
  • [26] A. Pełczyński, Banach spaces on which every unconditionally converging operator is weakly compact, Bull. Acad. Polon. Sci. 10 (1962), 641-648.
  • [27] A. Pełczyński, Banach Spaces of Analytic Functions and Absolutely Summing Operators, CBMS Regional Conf. Ser. in Math. 30, Amer. Math. Soc., Providence, R.I., 1977.
  • [28] R. M. Range, Holomorphic Functions and Integral Representations in Several Complex Variables, Springer, New York, 1986.
  • [29] P. Wojtaszczyk, On weakly compact operators from some uniform algebras, Studia Math. 64 (1979), 105-116.
  • [30] P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge Univ. Press, New York, 1991.
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