Approximation algorithm for an infinite-dimensional operator equation XL−BX=C

We study an infinite-dimensional operator equation XL–BX=C in a separable Hilbert space. The equation arises in the stabilization study of general linear parabolic systems, where the operatorsL, B, and C are coefficient operators describing a feedback control system. The solution to the stabilization naturally leads to an approximation problem of the operator equation. In this paper we propose a concrete algorithm for the approximation with the prescribed convergence rate when the closed operatorL is self-adjoint or more generally a spectral operator with compact resolvent.The original version of the paper was written while the author was with the Department of Mathematics, Faculty of Engineering, Kumamoto University, Kumamoto 860, Japan.