Spectral properties of non-selfadjoint Sturm-Liouville operator with operator coefficient

In this paper, we consider the Sturm-Liouville operator L generated in L2(R+,H) by the differential expressionL(Y)=−Y″+Q(x)Y,00. Here H is a separable Hilbert space, B(H) denotes the space of bounded operators in H and L2(R+,H) denotes the space of square-integrable, strongly-measurable vector-valued functions defined on (0,∞). In particular, we find some special solutions of the equation L(Y)=λ2Y including Jost solution, then investigate the point spectrum of L under certain conditions on Q(x). We obtain the resolvent of L, if Q(x) is quasi-selfadjoint i.e. there exists P∈B(H) such that P−1∈B(H),P is positive and Q⁎(x)=PQ(x)P−1 for every x>0. We also show that L has a finite number of spectral singularities.