A kernel theorem on the space [𝐻_{𝜇}×𝐴;𝐵

Abstract

Recently, we introduced a space [ H μ ( A ) ; B ] [{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 μ × A ; B ] [{H_\mu } \times A;B] , it can be uniquely represented by an element of [ H μ ( A ) ; B ] [{H_\mu }(A);B] and hence of [ H μ ; [ A ; B ] ] [{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 × V {D_{{R^n}}} \times V where V is a Fréchet space. His work is based on the facts that the space D R n {D_{{R^n}}} is an inductive limit space and the convolution product is well defined in D K j {D_{{K_j}}} . This is not possible here since the space H μ ( A ) {H_\mu }(A) is not an inductive limit space. Furthermore, D ( A ) D(A) is not dense in H μ ( A ) {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.