Asymptotics of the spectrum of Douglis-Nirenberg elliptic operators on a compact manifold

Pseudo-differential systems on closed manifolds, elliptic in the sense of DOUGLIS and NIRENBERG, are considered. It is proved that the system is similar to an diagonal operator up to an operator of order -infinity, and the similarity transformation preserves ellipticity and parameter-ellipticity. The similarity transformation may be chosen so that it preserves even self-adjointness up to an operator of order less than the lowest order of the diagonal entry. These results are applied to prove the eigenvalue asymptotics with the sharp estimate of the remainder in the self-adjoint case as well as a rough asymptotics in the non-self-adjoint case. Another application is the eigenvalue asymptotics with the sharp estimate of the remainder for general self-adjoint elliptic boundary value problems with the spectral parameter appearing linearly in boundary conditions.