A RESOLVENT COMPUTATION RELATED TO COMPLETENESS FOR COMPACT OPERATORS
Sjoerd M. Verduyn Lunel · WORLD SCIENTIFIC eBooks · 1995
In a recent paper [4] we have studied the problem whether the period map of a periodic delay equation has a complete span of eigenvectors and generalized eigenvectors. There is an abstract theory (see [4] and also [3]) that can be used to verify whether the eigenvectors and generalized eigenvectors corresponding to the nonzero spectrum of a compact operator from a given class of operators are complete. To use the abstract results one needs good estimates for the resolvent operator near infinity and to compute the resolvent explicitly one often has to solve a boundary value problem. In this paper we first give an abstract theorem and then we discuss some explicit examples. 1. A Result about Completeness Let H be a complex Hilbert space and let T : H ! H be a compact operator. Let E T denote the span of the eigenvectors and generalized eigenvectors corresponding to the nonzero eigenvalues of T . If E T is dense we call the system of eigenvectors and generalized eigenvectors complete....