Exact and approximate solutions of some operator equations based on the Cayley transform

We consider the operator equation SX = Sigma(j=1)(M) UjXVj = Y where {U-j}, {V-j} are some commutative sets of operators but in general {U-j} need not commute with {V-j}. Particular cases of this equation are the Sylvester and Ljapunov equations. We give a new representation and an approximation of the solution which is suitable to perform it algorithmically. Error estimates are given which show exponential covergence for bounded operators and polynomial convergence for unbounded ones. Based on these considerations we construct an iterative process and give an existence theorem for the operator equation Z(2) + A(1)Z + A(2) = 0, arising for example when solving an abstract second order differential equation with non-commutative coefficients. (C) 1998 Published by Elsevier Science Inc. All rights reserved.