Local spectral radii and Collatz-Wielandt numbers of monic operator polynomials with nonnegative coefficients

Operator polynomials L(λ) = λlI − λl−1Al−1 − … − λA1 − A0 are considered, where A0, …, Al−1 are nonnegative operators in a Banach space x with normal cone x+. For x x+ we define the local spectral radius rL(x) and the lower and upper Collatz-Wielandt numbers and , respectively, of x with respect to L. We characterize these quantities with the help of corresponding quantities with respect to the first companion operator belonging to L and the operator function S(λ) = Al−1 + λ−1Al−2 + … + λ−l+1A0. Many properties known in the linear case l = 1 have generalizations to the case l> 1; e.g., is true for all x x+. From these local results we obtain results for the global spectral radius r(L), which were proved earlier under more restrictive conditions.