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

✍ Scribed by K.-H. Förster; B. Nagy

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 666 KB
• Volume: 268
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/s0024-3795(97)89323-8

✦ Synopsis

Operator polynomials L(A)= htI-A I 1A l_ 1 ..... hA 1 -A o are considered, where A0,..., A t_l are nonnegative operators in a Banach space ~ with normal cone ~+. For x G ~-o~+ we define the local spectral radius rL(x) and the lower and upper Collatz-Wielandt numbers rL(x) and rL(x), 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(A) = A I 1 + A-1AI-2 + "'" +A-t+IA0 • Many properties known in the linear case 1 = 1 have generalizations to the case 1 > 1; e.g., rL(x) ~ rL(x) ~ ?L(X) is true for all x ~ Y+. From these local results we obtain results for the global spectral radius r(L), which were proved earlier under more restrictive conditions.