AN INDEX THEOREM FOR MONOTONE MATRIX-VALUED FUNCTIONS

The main result of this paper is the following index theorem, which is closely related to oscillation theorems on linear selfadjoint differe ntial systems such as results by M. Morse. Let real m x m-matrices R(1 ), R(2), X, U be given, which satisfy R(1)R(2)(T)=R(2)R(1)(T), X(T)U=U (T)X, rank (R(1), R(2)) = rank(X(T), U-T) = m.Moreover, assume that X( t), U(t) are real m x m-matrix-valued functions on some interval J = [ -epsilon, epsilon], epsilon > 0, such that X(T)(t)U(t)=U-T (t)X(t) on J, T X(t)-->X and U(t)-->U as t-->0, X(t) is invertible for t epsilon J\{0}, and such that U(t)X(-1)(t) is decreasing on J\{0}, and define M (t) = R(1)R(2)(T) + R(2)U(t)X(-1)(t)R(2)(T), Lambda(t) = R(1)X(t) R(2) U(t), Lambda = R(1)X + R(2)U. Then ind M(O+), ind M(0-), and def Lambd a(0+) exist and ind M(0+) - ind M(0-) = def Lambda - def Lambda(0+) - def X, where ind denotes the index (the number of negative eigenvalues ) and def denotes the defect (the dimension of the kernel) of a matrix . The basic tool for the proof of this result consists of a theorem on the rank of a certain product of matrices, so that this rank theorem is the key result of the present paper.