Some Krein spaces of analytic functions and an inverse scattering problem
Given a bounded self-adjoint operator A on a Hilbert space, one can define the following Hermitian forms on the range of A: \[ <Ax,Ay>_ A=<(A\quad *A)^{1/2}x,y>\text{ and } [Ax,Ay]_ A=<Ax,y>. \] Letting \({\mathcal K}\) denote the completion of range A in the topology induced by \(<\cdot,\cdot >_ A\), the author of the article under review shows that (\({\mathcal K},[\cdot,]_ A)\) is a Krein space. Moreover, one of the main results of the paper (Theorem 1) is that if the operator A is defined on the Hilbert space \(H^ 2_ n\) (the Hardy space of \(C^ n\)-valued square summable functions on the boundary of \(\Delta\)- either the open unit disc or the upper half-plane) by \((Af)(t)=X(t)J(PX\) *f)(t), where \(X \in H^{\infty}_{nm}\) (the \(n\times m\) matrices with \(H^{\infty}\) entries), \(P:L^ 2_ m\to H^ 2_ m\) is orthogonal projection and J is an operator on \(H^ 2_ m\) satisfying \(J=J^*=J^{-1}\), then the function \(k(x,y)=X(y)JX^*(x)/\rho(x,y)\) (for the cases of the circle and of the real line respectively, \(\rho(x,y)=1-yx^*\) and \(\rho(x,y)=- 2\pi i(y-x^*))\) is a reproducing kernel for the Krein space (\({\mathcal K},[\cdot,\cdot]_ A)\). (Recall that the function \(k(x,y)\) on \(\Delta\times \Delta\) with values in the \(n\times n\) matrices is a reproducing kernel provided that, for every \(f\in {\mathcal K}\), \(x\in \Delta\), and \(z\in {\mathbb{C}}^ n,\) the function \(k_ xz:y\mapsto k(x,y)z\) is in \({\mathcal K}\) and \([f,k(x,\cdot)z]=z^*f(x)\).) The author extends the analysis to include all X of bounded type. For simplicity, denote this reproducing kernel Krein space by \({\mathcal K}(X).\) Reproducing kernels of this type arise in the solution of the so-called `lossless inverse scattering problem' of network theory which is here adapted by the author to a Krein space setting. Specifically, he defines the `inverse scattering problem' for the function X (or more precisely for \(X(y)JX^*(x)/\rho(x,y))\) to consist of finding \(m\times m\) matrix valued functions, \(\theta\), of bounded type for which the map \(f\mapsto XJf\) is a contraction between the (reproducing kernel) Krein spaces \({\mathcal K}([I_ m\theta])\) and \({\mathcal K}(X)\). The main result presented in this context is Theorem 3 which applies to X in \(H^{\infty}_{nm}\) and explicitly produces a class of Krein spaces whose reproducing kernels are of the form \((J-\theta(y)J\theta^*(x))/\rho(x,y)\) where \(\theta\) is a solution to the inverse scattering problem for X.
scientific article; zbMATH DE number 4040565