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Transactions of the American Mathematical Society

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Convolution operators on $ G$-holomorphic functions in infinite dimensions

Authors: Philip J. Boland and Seán Dineen
Journal: Trans. Amer. Math. Soc. 190 (1974), 313-323
MSC: Primary 46G20; Secondary 47B37
MathSciNet review: 0407599
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Abstract: For a complex vector space E, let $ {H_G}(E)$ denote the space of G (Gateaux)-holomorphic functions on $ E\;(f:E \to C$ is G-holomorphic if the restriction of f to every finite dimensional subspace of E is holomorphic in the usual sense). The most natural topology on $ {H_G}(E)$ is that of uniform convergence on finite dimensional compact subsets of E. A convolution operator A on $ {H_G}(E)$ is a continuous linear mapping $ A:{H_G}(E) \to {H_G}(E)$ such that A commutes with translations. The concept of a convolution operator generalizes that of a differential operator with constant coefficients. We prove that if A is a convolution operator on $ {H_G}(E)$, then the kernel of A is the closed linear span of the exponential polynomials contained in the kernel. In addition, we show that any nonzero convolution operator on $ {H_G}(E)$ is a surjective mapping.

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  • [1] J. A. Barroso, Topologia em espaços de aplicações holomorfas entre espaços localmente convexos, An. Acad. Brasil. Ci. 43 (1971), 527-546. MR 0308760 (46:7874)
  • [2] P. J. Boland, Espaces ponderés de fonctions entières et de fonctions entières nucléaires sur un espace de Banach, C. R. Acad. Sci. Paris 75 (1972), 587-590. MR 0407598 (53:11370)
  • [3] -, Some spaces of entire and nuclearly entire functions on a Banach space, J. Reine Angew. Math. (to appear in two parts).
  • [4] S. Dineen, Holomorphic functions on locally convex topological vector spaces, Ann. Inst. Fourier (Grenoble), 23 (1973), 19-54. MR 0500153 (58:17843)
  • [5] C. P. Gupta, Convolution operators and holomorphic mappings on a Banach space, Séminaire d'Analyse Moderne, no. 2, Université de Sherbrooke.
  • [6] B. Malgrange, Existence et approximation des solutions des équations aux dérivées partielles et des équations des convolutions, Ann. Inst. Fourier (Grenoble) 6 (1955-56), 271-355. MR 19, 280. MR 0086990 (19:280a)
  • [7] Leopoldo Nachbin, Topology on spaces of holomorphic mappings, Ergebnisse der Math. und ihrer Grenzgebiete, Band 47, Springer-Verlag, New York, 1969. MR 40 #7787. MR 0254579 (40:7787)
  • [8] B. A. Taylor, Some locally convex spaces of entire functions, Proc. Sympos. Pure Math., Vol. 11, Entire Functions and Related Parts Analysis, 1968. MR 0240329 (39:1678)

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Keywords: Convolution operator, G (Gateaux)-holomorphic function, weighted space of G (Gateaux)-holomorphic functions, exponential-polynomial, Borel transform
Article copyright: © Copyright 1974 American Mathematical Society

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