## 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.

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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