Realizability Conditions for Time-Varying and Time-Invariant Hilbert Ports

Abstract

Let H be a complex Hilbert space, $R^n $n-dimensional real Euclidean space, and C the complex plane. $\mathcal{D}( H ) = \mathcal{D}_{R^n } ( H )$ denotes the space of infinitely differentiable functions from $R^n $ into H of compact support supplied with the customary Schwartz topology. Let $L_2 ( H )$ be the Hilbert space of H-valued quadratically integrable functions on $R^n $. Also, let $\mathcal{D}_{L_2 } ( H )$ be the space of functions $\phi $ from $R^n $ into H such that $\phi ^{( k )} \in L_2 ( H )$ for every derivative $\phi ^{( k )} $ of $\phi $. Finally, let $[ {U;V} ]$ denote the space of continuous linear mappings from U into V where U and V are topological linear spaces. The following two statements are equivalent. (i) $\mathfrak{M}$ is a linear scatter-semipassive mapping on $\mathcal{D}( H )$. (ii) $\mathfrak{M}$ has a composition representation $\mathfrak{M} = f \cdot $ where $f \in [ {\mathcal{D}_{R^{2n} } ( H );H} ]$, $\mathfrak{M}$ maps $\mathcal{D}( H )$ into $L_2 ( H )$, and $\iota - \hat f \circ f$ is a nonnegative kernel on $\mathcal{D}( H )$. Here, $\iota $ is the kernel of the composition representation for the identity operator on $\mathcal{D}( H )$. The identification between $[ {\mathcal{D}_{R^{2n} } ( H );H} ]$ and $[ {\mathcal{D}_{R^{2n} } ( C );[ {H;H} ]} ]$ allows us to define $\hat f$ by $\langle \hat f,\chi \rangle = \langle f,\bar {\chi^ \vee } \rangle ^\prime $, where $\chi \in \mathcal{D}_{R^{2n} } ( C )$, $\bar {\chi^ \vee } ( t,x ) = \bar \chi ( x,t )$, $t \in R^n $, $x \in R^n $. (The bar denotes the complex conjugate, and the prime denotes the adjoint operator.) Also, $\hat f \circ f$ is the Volterra product of $\hat f$ and f. This extends a result of R. Meidan who considered the case where $H = C$. A strengthening of this result in the special case where $\mathfrak{M}$ is translation-invariant (i.e., time-invariant when $R^n = R^1 $) is also achieved.