The index of triangular operator matrices

For any triangular operator matrix acting in a direct sum of complex Banach spaces, the order of a pole of the resolvent (i.e. the index) is determine d as a function of the coefficients in the Laurent series for all the (reso lvents of the) operators on the diagonal and of the operators below the dia gonal. This result is then applied to the case of certain nonnegative opera tors in Banach lattices. We show how simply these results imply the Rothblu m Index Theorem (1975) for nonnegative matrices. Finally, examples for calc ulating the index are presented.