Existence of a low rank or H-matrix approximant to the solution of a Sylvester equation

We consider the Sylvester equation AX - AB + C = 0 where the matrix C is an element of C-nxm is of low rank and the spectra of A is an element of C-n x n and B is an element of C-m x m are separated by a line. We prove that the singular values of the solution X decay exponentially, that means for any epsilon is an element of (0, 1) there exists a matrix (X) over tilde of rank k = O(log(l/epsilon)) such that \\X - (X) over tilde \\(2) less than or equal to epsilon\\X\\(2). As a generalization we prove that if A,B. C are hierarchical matrices then the solution X can be approximated by the hierarchical matrix formal described in Hackbusch (Computing 2000; 62:89-108). The blockwise rank of the approximation is again proportional to log(l/epsilon). Copyright (C) 2004 John Wiley Sons, Ltd.