A Kernel Theorem on the Space [ H μ × A; B

Abstract

Recently, we introduced a space [Hfi(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 [Hf, x A ; B], it can be uniquely represented by an element of [H^A) ; B] and hence of [H^ ; [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 Dgn x V where F is a Frechet space. His work is based on the facts that the space D$n is an inductive limit space and the convolution product is well defined in Dr . This is not possible here since the space Hn(A) is not an inductive limit space. Furthermore, D(A) is not dense in Hf,(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 .