Superadditive Perron–Frobenius theory. (English) Zbl 1199.31022
Bakry, Dominique (ed.) et al., Current trends in potential theory. Proceedings of the 2nd IMAR workshop on potential theory, Bucharest, Romania, September 9–29, 2002, and the potential theory conference, Bucharest, Romania, September 23–27, 2003. Bucharest: Theta (ISBN 973-85432-6-6). Theta Series in Advanced Mathematics 4, 85-98 (2005).
There are generalizations of the Perron – Frobenius theory to proper cones, to infinite dimensions and to nonlinear maps; see B.-S. Tam [Taiwanese J. Math. 5, No. 2, 207–277 (2001; Zbl 0990.15009)]. The author considers positively homogeneous and superadditive maps \(\Lambda\), which act continuously on a finite dimensional proper subcone \(\mathbb P\) of a Euclidean space. He investigates the existence of eigenvectors at the boundary of the cone and uses Gateaux derivatives of \(\Lambda\) at the boundary, \(\partial P\), of the cone to detect nonlinearities and to turn them efficiently into estimates of Collatz–Wielandt numbers.
For the entire collection see [Zbl 1108.31001].
For the entire collection see [Zbl 1108.31001].
Reviewer: Mohammad Sal Moslehian (Mashhad)
MSC:
| 31C25 | Dirichlet forms |
| 47H07 | Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces |
| 15B48 | Positive matrices and their generalizations; cones of matrices |
