Exact and approximate solutions of some operator equations based on the Cayley transform
The operator equation \[ \sum^M_{j=1} U_jXV_j= Y \] is studied in the paper. Here \(\{U_j\}\) and \(\{V_j\}\) are commuting sets of linear bounded Banach space operators; however, in general \(\{U_j\}\) need not commute with \(\{V_j\}\). Assuming that the operator equation has a unique solution \(X\) for every \(Y\), and under the additional hypothesis that the operators \(\{U_j\}\) and \(\{V_j\}\) are strongly positive, constructive approximations to the solution of the operator equation and their error estimates are given. The error estimate for the \(N\)th approximant \(X_N\) is of the form \[ \|X- X_M\|\leq c\|Y\|N^{2M-1} q^N, \] where \(c>0\) and \(0<q<1\) are constants independent of \(N\). The method of proof is based on explicit integral representations and on norm estimates for powers of the Cayley transform of the operators \(\{U_j\}\) and \(\{V_j\}\). A weaker version of the results is proved for the case when the operators \(\{U_j\}\) and \(\{V_j\}\) are allowed to be unbounded. The general results are further specialized to several particular situations, including the Lyapunov equation \(UX+ XV= Y\), the quadratic operator equation \(Z^2+ A_1Z+ A_2\) for operators in Hilbert space, and an abstract second-order initial value problem with non-commuting coefficients.
scientific article; zbMATH DE number 1370158
scientific article