Realizability Theory of Continuous Linear Operators on Groups

Abstract

Let G and H be a certain type of locally compact (abelian) group. $D(G)$ denotes the space of regular functions with compact support on G and $D'(G)$ is the corresponding space of distributions. Linear mappings from $D(G)$ into $D'(G)(D'(H))$ are the subject of our investigations. We have carried out some systems-theoretic investigation in a distributional setting where distributions are defined on groups. Such investigations in (Schwartz’s) distributional setting have been carried out by several authors. We have chosen the distribution theory on groups as developed by F. Bruhat, K. Maurin and G. I. Kac. This choice is motivated by the existence of Bruhat’s kernel theorem and the nuclearity of the space $D(G)$. Properties such as continuity, regularity, translation-invariance, causality, semipassivity and passivity (a certain positivity property) are imposed on the linear mappings and their effects are studied. Representations of continuous linear mappings from $D(G)$ into $D'(H)$ which are regular into $C(H)$, of continuous linear and translation-invariant mappings from $D(G)$ into $D'(G)$ and of linear and scatter-semipassive mappings from $D(G)$ into $D'(G)$ are obtained. We establish one-to-one correspondence between contractions in $L_2 (G)$ and linear and scatter-semipassive mappings from $D(G)$ into $D'(G)$. We also show that causality and semipassivity imply passivity in the scattering formalism and passivity implies causality in the immittance formalism. A characterization of linear, scatter-semipassive, and real mappings in terms of a positivity criterion is established.