## ASSOCIATED FUNCTIONS OF NON-SELFADJOINT STURM-LIOUVILLE OPERATOR WITH OPERATOR COEFFICIENT

## ASSOCIATED FUNCTIONS OF NON-SELFADJOINT STURM-LIOUVILLE OPERATOR WITH OPERATOR COEFFICIENT

*G. Mutlu*

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## 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 Sturm- Liouville 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 correspond- ing to the eigenvalues belong to L2 (R+,H) while the associated functions corresponding to the spectral singularities do not.

## Keywords

Sturm-Liouville operator equation, associated functions, operator coefficient, non-selfadjoint operators.