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Michael Orlitzky (michaelorlitzky.com) Abstract: Let K be a closed convex cone with dual Kstar in a finitedimensional 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 PerronFrobenius theory. We say that L is a Zoperator on K if <L(x), s> <= 0 for all (x, s) in K x Kstar such that <x,s> = 0. The Zoperators are generalizations of Zmatrices (whose offdiagonal elements are nonpositive) and they arise in dynamical systems, economics, game theory, and elsewhere. We connect the positive and Zoperators. This extends the work of Schneider, Vidyasagar, and Tam on proper cones, and reveals some interesting similarities between the two families. Keywords: Positive operator, Zoperator, Zmatrix, Lyapunovlike, exponentiallypositive, dynamical systems Category 1: Convex and Nonsmooth Optimization (Convex Optimization ) Category 2: Linear, Cone and Semidefinite Programming (Other ) Citation: Michael J. Orlitzky. Positive and Zoperators on closed convex cones. Electronic Journal of Linear Algebra, 34:444458, 2018. Download: Entry Submitted: 09/26/2016 Modify/Update this entry  
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