Abstract
Given two complex Banach spaces E and F, convolution operators “with scalar coefficients” are characterized among all convolution operators on the space H N b ( E ′ ; F ) {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 E x p ( E ; F ′ ) 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 N b ( E ′ ; F ) {H_{Nb}}(E’;F) . THEOREM 2. Solutions of homogeneous convolution equations of scalar type can be approximated in H N b ( E ′ ; F ) {H_{Nb}}(E’;F) by exponential-polynomial solutions.
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