ON MODULES OF EIGENVALUES FOR NONSELF-ADJOINT AGMON-DOUGLIS-NIRENBERG ELLIPTIC BOUNDARY-PROBLEMS WITH A PARAMETER

In this note we describe some generalizations of theorems obtained in [1, 2] concerning spectral asymptotics. We consider operators of two types: 1) matrix Douglis-Nirenberg elliptic operators; 2) operators polynomially depending on a spectral parameter. In the author's opinion, the most convenient approach to operators of the second type consists in their reduction to general operators of the first type. To obtain appropriate results for operators of the first type on a manifold without boundary, it is quite sufficient to use elementary reductions indicated, e.g., in [3, Section 4.31; see Section 1 below. But the case of a manifold with boundary is technically more complicated. Our main purpose in this note is to consider just this case. In [4] we used an analogous approach for the description of those subspaces in Sobolev spaces on a manifold with boundary in which we have completeness or multiple completeness of root functions (i.e., generalized eigenfunctions). We indicate a rather simple way of obtaining the key result from [4]: see the remark at the end of the present note.