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Positive and Z-operators on closed convex cones

Michael Orlitzky (michael***at***orlitzky.com)

Abstract: Let K be a closed convex cone with dual K-star in a finite-dimensional real Hilbert space V. A positive operator on K is a linear operator L on V such that L(K) is a subset of K. Positive operators generalize the nonnegative matrices and are essential to the Perron-Frobenius theory. We say that L is a Z-operator on K if <L(x), s> <= 0 for all (x, s) in K x K-star such that <x,s> = 0. The Z-operators are generalizations of Z-matrices (whose off-diagonal elements are nonpositive) and they arise in dynamical systems, economics, game theory, and elsewhere. We connect the positive and Z-operators. This extends the work of Schneider, Vidyasagar, and Tam on proper cones, and reveals some interesting similarities between the two families.

Keywords: Positive operator, Z-operator, Z-matrix, Lyapunov-like, exponentially-positive, dynamical systems

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )

Category 2: Linear, Cone and Semidefinite Programming (Other )

Citation: Michael J. Orlitzky. Positive and Z-operators on closed convex cones. Electronic Journal of Linear Algebra, 34:444-458, 2018.


Entry Submitted: 09/26/2016
Entry Accepted: 09/26/2016
Entry Last Modified: 10/08/2018

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