Schur complements and state space realizations
Given a \(2\times2\) operator matrix \[ S= \left[\begin{matrix} A & B\\ \mathbb C & D \end{matrix}\right]:X\dot{+}Y\to\widehat{X} \dot{+}\widehat{Y}, \] where \(X,Y,\widehat{X}\) and \(\widehat{Y}\) are complex Banach spaces, the first and the second Schur complements in \(S\) are defined, respectively, by \(W_1(S)=D-CA^{-1}B:Y\to\widehat{Y}\) if \(A\) is invertible, and by \(W_2(S)=A-BD^{-1}C:X\to\widehat{X}\) if \(D\) is invertible. Several operations on Schur complements related to state space realizations of transfer functions of linear time invariant systems are introduced and studied. First the authors study two operations that preserve the first Schur complement and relate to state space similarity and dilation. Schur coupling, equivalence after extension, and matricial coupling are also studied. Two operators \(W_1\) and \(W_2\) are called Schur coupled if there exists a \(2\times2\) matrix \(S\) such that \(W_1(S)=W_1\) and \(W_2(S)=W_2\). This implies that \(W_1\) and \(W_2\) are equivalent after (two-sided) extension, that is, there exist Banach spaces \(X\) and \(Y\) and invertible operators \(E\) and \(F\) such that \(W_1\dot{+}I_X=E(W_2\dot{+}I_Y)F\). The latter identity is equivalent to the matricial coupling of \(W_1\) and \(W_2\). It is shown that two operators that are equivalent after one-sided extension are Schur coupled. Further, for \(2\times2\) operator matrices an ``\(S\)-product'' \(\diamondsuit\) and an ``\(S\)-inverse'' \(^{-\diamondsuit}\) are introduced in such a way that \[ W_1(S_1\diamondsuit S_2)=W_1(S_1) W_1(S_2),\quad W_1(S^{-\diamondsuit})= W_1(S)^{-1}. \] The \(S\)-product is associative, the \(S\)-inverse is an involution and, modulo a simple transformation, \((S_1\diamondsuit S_2)^{-\diamondsuit}\) can be identified with \(S_2^{-\diamondsuit}\diamondsuit S_1^{-\diamondsuit}\). Finally, a factorization of the first Schur complement related to pairs of matching invariant subspaces is constructed and a certain kind of its generalization is studied.
scientific article; zbMATH DE number 2160189