Weylʹs theorem in several variables

In this note we consider Weylʹs theorem and Browderʹs theorem in several variables. The main result is as follows. Let T be a doubly commuting n-tuple of hyponormal operators acting on a complex Hilbert space. If T has the quasitriangular property, i.e., the dimension of the left cohomology for the Koszul complex Λ ( T − λ ) is greater than or equal to the dimension of the right cohomology for Λ ( T − λ ) for all λ ∈ C n , then ‘Weylʹs theorem’ holds for T, i.e., the complement in the Taylor spectrum of the Taylor Weyl spectrum coincides with the isolated joint eigenvalues of finite multiplicity.