Completeness of root vectors for an Agmon-Douglis-Nirenberg elliptic problem with an indefinite weight in $L_p$ spaces

Abstract

This paper is devoted to the systematic investigation of the completeness of root vectors (generalized eigenvectors) of a non-selfadjoint Agmon-Douglis-Nirenberg (ADN) elliptic boundary problem with an indefinite weight matrix. As it is known the realization of a ADN elliptic problem (or simply a correctly posed general elliptic boundary problem in which no restrictions are imposed on the order of some boundary operators) is not in general densely defined. This amounts to some tremendous difficulties in the derivation of the completeness of the root vectors for the corresponding spectral problem. However under some circumstances the completeness of root vectors may be obtained in some dense subsets of functions in Sobolev spaces. These issues are addressed in the present work for a class of non-selfadjoint ADN elliptic boundary value problems with an indefinite weight matrix-function for which we establish new completeness results for the corresponding root vectors in appropriate $L_{p}$ Sobolev spaces.