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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Convolution operators on $G$-holomorphic functions in infinite dimensions
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by Philip J. Boland and Seán Dineen PDF
Trans. Amer. Math. Soc. 190 (1974), 313-323 Request permission

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.
References
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Additional Information
  • © Copyright 1974 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 190 (1974), 313-323
  • MSC: Primary 46G20; Secondary 47B37
  • DOI: https://doi.org/10.1090/S0002-9947-1974-0407599-9
  • MathSciNet review: 0407599