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 t he Sylvester and Ljapunov equations. We give a new representation and an approximation of the solution which is suitable to perform it algor ithmically. Error estimates are given which show exponential covergenc e for bounded operators and polynomial convergence for unbounded ones. Based on these considerations we construct an iterative process and g ive an existence theorem for the operator equation Z(2) + A(1)Z + A(2) = 0, arising for example when solving an abstract second order differ ential equation with non-commutative coefficients. (C) 1998 Published by Elsevier Science Inc. All rights reserved.