ON GLOBAL REPRESENTATIONS OF THE SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS AS A PRODUCT OF EXPONENTIALS J. WEI AND E. NORMAN 1. Introduction. In this paper we shall consider solutions of the equation dUit) (1) —^- = Ait)Uit), 17(0) = /, at where A and U are linear operators, and / is the identity operator. Our results will be applicable when the operator A it) can be written as m ■^ it) — S aiit)Xi, m finite, ;=i where the a,-(i) are scalar functions of t, and the operators Xf are independent of t. It is further required that the Lie algebra £ gener- ated by the X< under the commutator product [X{, X¡]=XíXj —XjXi be of finite dimension /. The above is, of course, always true if A (and U) are finite matrix operators. In 1954, W. Magnus [4] proved that if Xu X2, • • • , X¡ is a basis for £, then the solution of (1) can be expressed in the form /7(f) = exp(2í=i giit)Xi). 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 2X2 matrices a global version of Magnus' result cannot be obtained without severe restric- tions on Ait). We will show that if U is a solution of (1), it can be represented in the form (2) U(t)=n«*P (!<(<)*<). 1=1 This representation is global for all solvable Lie algebras, and for any real 2X2 system of equations. The form (2) derives its principal utility from the fact that insight into the properties of t/(£) can be gained through a knowledge of the spectral properties of the individual operators X<. Since the Xi's are Received by the editors July 26, 1962 and, in revised form, January 17, 1963. 327 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use