Limit properties of monotone matrix functions

The basic objects in this paper are monotonically nondecreasing n×n matrix functions D(·) defined on some open interval ı=(a,b) of R and their limit values D(a) and D(b) at the endpoints a and b which are, in general, selfadjoint relations in Cn. Certain space decompositions induced by the matrix function D(·) are made explicit by means of the limit values D(a) and D(b). They are a consequence of operator inequalities involving these limit values and the notion of strictness (or definiteness) of monotonically nondecreasing matrix functions. This treatment provides a geometric approach to the square-integrability of solutions of definite canonical systems of differential equations.