Local spectral radii and Collatz-Wielandt numbers of monic operator polynomials with nonnegative coefficients
Operator polynomials \(L(\lambda)=\lambda^{l}I-\lambda^{l-1}A_{l-1}-\cdots-\lambda A_{1}-A_{0}\) are considered, where \(A_{0},\ldots,A_{l-1}\) are nonnegative operators in an ordered Banach space \({\mathcal H}\) with normal cone \({\mathcal H}_{+}\). For \(x\in{\mathcal H}_{+}\) local characteristics mentioned in the title are defined. These quantities are characterized with the help of corresponding quantities with respect to the first companion operator and the operator function \(S(\lambda) = A_{l-1}+\lambda^{-1}A_{l-2}+\cdots+\lambda^{-l+1}A_{0}\). Some properties known in the linear case \(l=1\) are generalized to the case \(l>1\). From these local results the authors obtain results for the global spectral radius \(r(L)\), which were proved earlier under more restrictive conditions.
scientific article; zbMATH DE number 1100486
scientific article