A CONE-THEORETIC APPROACH TO THE SPECTRAL THEORU OF POSITIVE LINEAR OPERATORS: THE FINITE-DIMENSIONAL CASE

Abstract

This is a review of a coherent body of knowlegde, which perhaps deserves the name of the geometric spectral theory of positive linear operators (in finite dimension), developed by this author and his co-author Hans Schncider (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 result are given. Several new new results are proved or announced, and some open problems are also mentioned.