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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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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