Associated functions of non-selfadjoint Sturm-Liouville operator with operator coefficient

Abstract

Sturm-Liouville operator equation with selfadjoint operator coefficent has been studied in detail. In this paper, we consider the Sturm-Liouville operator equation with non-selfadjoint operator coefficent. Namely, we examine the non-selfadjoint SturmLiouville operator L which is generated in L2(R+, H) by the differential expression L(Y ) = ?Y’’ + Q(x)Y, 0 < x < ?, with operator coefficient together with the boundary condition Y (0) = 0, where Q(x) is a non-selfadjoint, completely continuous operator in a separable Hilbert space H for each x ? (0, ?). We find the associated functions corresponding to the eigenvalues and spectral singularities of L. Moreover, we prove that the associated functions corresponding to the eigenvalues belong to L2 (R+, H) while the associated functions corresponding to the spectral singularities do not.