On the optimality for cascade connection of passive scattering systems and the best minorant outer function. (Q1607612)

A linear system \(\alpha =(X,U,V,A,B,C,D)\) is given by the equations \(x_{n+1}=Ax_n+Bu_n,\;v_n=Cx_n+Du_n,\;n\in {\mathbb N}\), where \(x_n,u_n,v_n\) belong to separable Hilbert spaces \(X,U,V\), respectively, and \(A,B,C,D\) are bounded linear operators between the corresponding pairs of spaces. Such a system \(\alpha\) is called a passive scattering system if the operator \(T=\left[\begin{smallmatrix} A & B \\ C & D \end{smallmatrix} \right]\) is a contraction. The transfer function \(\theta_{\alpha}(z)=D+zC(I-zA)^{-1}B\) of a passive scattering system \(\alpha\) is an analytic contractive operator-valued function on the unit disk. A passive scattering system \(\alpha =(X,U,V,A,B,C,D)\) is said to be optimal [see \textit{D. Z. Arov}, Sov. Math., Dokl. 20, 676--680 (1979); translation from Dokl. Akad. Nauk SSSR 247, 265--268 (1979; Zbl 0479.93020)] if for any other passive scattering system \(\alpha' =(X',U,V,A',B',C',D)\) with the same transfer function, and any finite sequence \(\{ u_k\}_{k=1,\dots,n}\subset U\), one has \(\| \sum_{k=0}^n A^k Bu_k\| \leq\| \sum_{k=0}^n {A'}^k B'u_k\|\), \(n\in {\mathbb N}\), \(u_k\in U.\) A passive scattering system \(\alpha =(X,U,V,A,B,C,D)\) is said to be *-optimal if its dual system \(\alpha_* =(X,V,U,A^*,C^*,B^*,D^*)\) is optimal. Let \(\alpha_k =(X_k,U_k,V_k,A_k,B_k,C_k,D_k),\;k=1,2\), be two linear systems satisfying \(V_1=U_2\). Then the system \(\alpha =(X_1\oplus X_2,U_1,V_2,A_1P_1+A_2P_2+B_2C_1P_1,B_1+B_2D_1,D_2C_1P_1+C_2P_2,D_2D_1)\) where \(P_k\) is the orthoprojector from \(X=X_1\oplus X_2\) onto \(X_k\;(k=1,2)\) is called a cascade connection of \(\alpha_1\) and \(\alpha_2\) and is written as \(\alpha=\alpha_2\alpha_1\). It is known that \(\theta_{\alpha}(z)=\theta_{\alpha_2}(z)\theta_{\alpha_1}(z)\). In the paper under review, the criteria for the optimality (resp., *-optimality) of the cascade connection \(\alpha=\alpha_2\alpha_1\) of two optimal (resp., *-optimal) passive scattering systems \(\alpha_1\) and \(\alpha_2\) are given in terms of the best minorant outer (resp., *-outer) functions of \(\alpha_1\), \(\alpha_2\) and \(\alpha\) [see \textit{B. Sz.-Nagy} and \textit{C. Foiaş}, Harmonic analysis of operators on Hilbert spaces, Budapest: Akadémiai Kiadó; Amsterdam-London: North-Holland (1970; Zbl 0201.45003)] for the notions of best minorant outer (resp., *-outer) functions).