Multiplicative Lower-Upper Triangular Decompositions of Operators

Abstract

In Chapter XX we dealt with the problem of additive LU-decompositions of operators with respect to a chain. In this chapter we are concerned with the more challenging problem of multiplicative LU-decomposition of certain operators. The theorems which are presented encompass classical results from Linear Algebra which show that, under certain conditions, a matrix can be represented as a product of a lower with an upper triangular matrix. Also, the possibility of factoring certain integral operators as the product of integral operators with lower and upper triangular kernel functions is treated as a special case of the general theory which we now develop.