A kernel theorem on the space $[H_ \mu \times A;B]$

Abstract:

Recently, we introduced a space $[{H_\mu }(A);B]$ which consists of Banach space-valued distributions for which the Hankel transformation is an automorphism (The Hankel transformation of a Banach space-valued generalized function, Proc. Amer. Math. Soc. 119 (1993), 153-163). One of the cornerstones in distribution theory is the kernel theorem of Schwartz which characterizes continuous bilinear functionals as kernel operators. The object of this paper is to prove a kernel theorem which states that for an arbitrary element of $[{H_\mu } \times A;B]$, it can be uniquely represented by an element of $[{H_\mu }(A);B]$ and hence of $[{H_\mu };[A;B]]$. This is motivated by a generalization of Zemanian (Realizability theory for continuous linear systems, Academic Press, New York, 1972) for the product space ${D_{{R^n}}} \times V$ where V is a Fréchet space. His work is based on the facts that the space ${D_{{R^n}}}$ is an inductive limit space and the convolution product is well defined in ${D_{{K_j}}}$. This is not possible here since the space ${H_\mu }(A)$ is not an inductive limit space. Furthermore, $D(A)$ is not dense in ${H_\mu }(A)$. To overcome this, it is necessary to apply some results from our aforementioned paper. We close this paper with some applications to integral transformations by a suitable choice of A.