A trace theorem and a linearization method for operator polynomials

Let p() = ⁿI - ^n-1 A₁ - A_n, where A_i are operators on a Banach space X with (1/i)^th power summable approximation numbers. Then the sum of the eigenvalues of the operator polynomial p() equals the trace of A₁. This is shown by a linearization method which is further used to generalize results on the completeness of the eigenvalues of p() under suitable conditions on the resolvent.