Eigenvalue Expansion of Nonsymmetric Linear Compact Operators in Hilbert Space

Abstract

For a symmetric linear compact resp. symmetric densely defined linear operator with compact inverse, expansion theorems in series of eigenvectors are known. The aim of the present paper is to generalize the known expansion theorems to the case of corresponding operators without the symmetry property. For this, we replace the set of orthonormal eigenvectors in the symmetric case by a set of biorthonormal eigenvectors resp. principal vectors in the case of simple eigenvalues resp. general eigenvalues. The results for the operators without the symmetry property are all new. Furthermore, if the operators are symmetric, the generalized results deliver the known expansions. As an application of the results for nonsymmetric operators with simple eigenvalues, we obtain a known expansion in a series of eigenfunctions for a non-selfadjoint Boundary Eigenvalue Problem with ordinary differential operator discussed in a book of Coddington/Levinson. But, we obtain a new result if the eigenvalues are general, that is, not necessarily simple. In addition, for a differential operator of 2nd order with constant coefficients, the eigenfunctions and Green’s function are explicitly determined. This result is also new, as far as the author is aware.