Let \(K\) be a proper cone in a vector space \(V\). A linear operator (matrix) \(A\) is called cone-preserving if \(AK\subseteq K\).
The paper is a review of the geometric spectral theory of positive linear operators (in finite dimension) developed by the author, Hans Schneider and S.F. Wu. This is a generalization of well-known Perron-Frobenius theory of non-negative matrices. The paper is an expanded version of an invited talk of the author at the International Conference on Mathematical Analysis and Its Applications held at National Yat-sen University in Taiwan in January 2000. It may serve as an introduction to the subject.
In Section 1 – Introduction – the classical Perron-Frobenius theory and some of its known generalizations are briefly described. In Section 2 combinatorial (graph-theoretical) properties of cone preserving matrices are studied. In Section 3 Collatz-Wielandt sets and numbers are introduced and used. Usefulness of the study of the core of a cone-preserving map \(A\), i.e., \(\bigcap_{i=1}^{\infty}A^iK\), is explained in Section 4. Section 5 is devoted to the investigation of invariant faces of a cone preserving map \(A\) and their relations to properties of \(A\). The concluding Section 6 contains some further results and remarks concerning the results of the previous sections.
The paper contains also some new results and open questions. The list of references (145 items) might be also very useful.