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Transactions of the American Mathematical Society

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



Convolution equations for vector-valued entire functions of nuclear bounded type

Author: Thomas A. W. Dwyer
Journal: Trans. Amer. Math. Soc. 217 (1976), 105-119
MSC: Primary 46G99; Secondary 32H15
MathSciNet review: 0487450
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Abstract: Given two complex Banach spaces E and F, convolution operators ``with scalar coefficients'' are characterized among all convolution operators on the space $ {H_{Nb}}(E';F)$ of entire mappings of bounded nuclear type of E' into F. The transposes of such operators are characterized as multiplication operators in the space $ Exp(E;F')$ of entire mappings of exponential type of E into F'. The division theorem for entire functions of exponential type of Malgrange and Gupta is then extended to the case when one factor is vector-valued. With this tool the following ``vector-valued'' existence and approximation theorems for convolution equations are proved: THEOREM 1. Nonzero convolution operators ``of scalar type'' are surjective on $ {H_{Nb}}(E';F)$. THEOREM 2. Solutions of homogeneous convolution equations of scalar type can be approximated in $ {H_{Nb}}(E';F)$ by exponential-polynomial solutions.

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Keywords: Infinite-dimensional holomorphy, convolution equations, vector-valued entire functions
Article copyright: © Copyright 1976 American Mathematical Society