On global representations of the solutions of linear differential equations as a product of exponentials

Abstract

where the ai(t) are scalar functions of t, and the operators Xi are independent of t. It is further required that the Lie algebra 2 generated by the Xi under the commutator product [Xi, Xj] = XiXj -XjXi be of finite dimension 1. The above is, of course, always true if A (and U) are finite matrix operators. In 1954, W. Miagnus [4] proved that if X1, X2, , Xi is a basis for ?, then the solution of (1) can be expressed in the form U(t) exp( Ei= gi(t)Xj). This representation of U holds, however, only in a neighborhood of the origin. It has been shown by J. Mariani and W. Magnus [3] that even in the case of 2 X 2 matrices a global version of Magnus' result cannot be obtained without severe restrictions on A (t). We will show that if U is a solution of (1), it can be represented in the form