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 syste m is similar to an diagonal operator up to an operator of order -infin ity, and the similarity transformation preserves ellipticity and param eter-ellipticity. The similarity transformation may be chosen so that it preserves even self-adjointness up to an operator of order less tha n the lowest order of the diagonal entry. These results are applied to prove the eigenvalue asymptotics with the sharp estimate of the remai nder in the self-adjoint case as well as a rough asymptotics in the no n-self-adjoint case. Another application is the eigenvalue asymptotics with the sharp estimate of the remainder for general self-adjoint ell iptic boundary value problems with the spectral parameter appearing li nearly in boundary conditions.