Cone-theoretic generalization of total positivity

TL;DR

This paper extends the theory of total positivity by defining generalized totally positive operators using cone preservation, analyzing their spectral properties, and providing new insights into classical results.

Contribution

It introduces the concept of generalized totally positive operators, explores their spectral properties, and offers new proofs and insights into classical total positivity results.

Findings

GSTP operators have positive, simple spectra

Eigenvectors of GSTP are localized in specific sets

Invariant cones of finite ranks can exist under certain conditions

Abstract

This paper is devoted to the generalization of the theory of total positivity. We say that a linear operator A in R^n is generalized totally positive (GTP), if its jth exterior power preserves a proper cone K_j in the corresponding space for every j = 1, ..., n. We also define generalized strictly totally positive (GSTP) operators. We prove that the spectrum of a GSTP operator is positive and simple, moreover, its eigenvectors are localized in special sets. The existence of invariant cones of finite ranks is shown under some additional conditions. Some new insights and alternative proofs of the well-known results of Gantmacher and Krein describing the properties of TP and STP matrices are presented.