Some properties of the spectral radius of a monic operator polynomial with nonnegative compact coefficients
Consider the monic operator polynomial \(L(\lambda)={\lambda}^ \ell - {\lambda}^{\ell-1}A_{\ell-1}-\cdots-\lambda A_ 1 - A_ 0\), where \(A_ 0, \dots ,A_{\ell-1}\) are linear non-negative compact operators acting on a Banach lattice. Assume that the spectral radius \(r\) of \(L\) is positive, and put \(S_ r= A_{\ell-1}+\cdots+r^{-\ell+2}A_ 1 +r^{-\ell+1}A_ 0\). The authors show that the negative compact operator \(S_ r\) and the polynomial \(L\) have the same spectral radius r and have similar spectral properties at \(r\) . For example, they prove that \(L^{-1}\) and the resolvent of \(S_ r\) have a pole at \(r\) of the same order, and that the generalized eigenspaces of \(S_ r\) and of the companion operator \(C_ L\) at \(r\) and the chain space of \(L\) corresponding to \(r\) have the same dimensions. They also show that every non-trivial \(L\)-chain corresponding to \(r\) is linearly independent. Recently, these and related results have been generalized to arbitrary monic operator polynomials with non-negative operator coefficients without a compactness condition on the coefficients [see \textit{R. T. Rau}, Integral Equations Oper. Theory 15, 479-495 (1992)].
linear non-negative compact operators acting on a Banach lattice
scientific article; zbMATH DE number 59722