9. Semi-Separable Operators and Completeness

In this chapter we specify further the results of Chaps. 4, 5 and 6 for semi-separable operators. The chapter consists of three sections (and two subsections). The first section deals with completeness results for a class of discrete semi-separable operators studied in Section V.3 of [29]. Such operators are quite different from the Leslie operators considered in Chap. 8. The theory developed in Chap. 4 is useful in studying completeness for these discrete semi-separable operators. The second section deals with completeness results for semi-separable integral operators. The results in Chapter IX of [32], together with the results developed in Chaps. 5 and 6, are used to derive completeness results for the semi-separable integral operators. The final section has a preparatory character, not directly related to semi-separable operators. We collect some results about the fundamental solution of an ordinary differential equation, and we present an explicit resolvent formula for a class of integral operators and related Volterra operators which will play a role in the next chapter and in Chap. 11.