Decompositions of a Hilbert Space and Factorization of a W–A Determinant

The dispersion function $\Lambda (\lambda )$ which occurs in linear transport theory can be introduced as the W-A determinant of a certain pair of operators $B_1 $, $B_2 $ defined in $L^2 [ - 1,1]$. Each of the two operators is reduced by a complementary pair of subspaces of $L^2 [ - 1,1]$. In this paper the factorization $\Lambda (\lambda ) = X(\lambda )X( - \lambda )$ is shown to correspond with a factorization of the operator $(VB_2 V^{ - 1} - \lambda E) \cdot (B_1 - \lambda E)^{ - 1} $ into the product of two operators with determinants $X( \pm \lambda )$. Here $V$ is an automorphism of $L^2 [ - 1,1]$ which is defined in terms of the projections associated with the two pairs of subspaces. The results are brought into a general setting.