A cone-theoretic approach to the spectral theory of positive linear operators: The finite-dimensional case

Abstract

This is a review of a coherent body of knowledge, which perhaps deserves the name of the geometric spectral theory of positive linear operators (in finite dimensions), developed by this author and his co-author Hans Schneider (or S.F. Wu) over the past decade. The following topics are covered, besides others: combinatorial spectral theory of nonnegative matrices, Collatz-Wielandt sets (or numbers) associated with a cone-preserving map, distinguished eigenvalues, cone-solvability theorems, the peripheral spectrum and the core, the invariant faces, the spectral pairs, and an extension of the Rothblum Index Theorem. Some new insights, alternative proofs, extensions or applications of known results are given. Several new results are proved or announced, and some open problems are also mentioned.