Cone-theoretic generalization of total positivity

Linear Algebra and Its Applications, 1992

If K, is a proper cone in R"I and K, is a proper cone in R"z, then, as is well known, the set *(K,, K,), which consists of all ne x nl real matrices which take K, into K,, forms a proper cone in the space R"2, "I. In this paper a study of this cone is made, with particular emphasis on its faces and duality operator. A face of r(K,, K,) is called simple if it is composed of all matrices in t( K,, K2) which take some fixed face of K, into some fixed face of K,. Maximal faces of r(K,, K,) are characterized as a particular kind of simple faces. Relations between the duality operator of X( K,, K,) and those of K, and K, are obtained. Among many other results, it is proved that d r(K,, K2), the duality operator of r( K,, K,), is injective if and only if dK, is injective and each face of K( K,, K,) is an intersection of simple faces. Two open questions are posed.

2017

We prove that the only entrywise transforms of rectangular matrices which preserve total positivity or total non-negativity are either constant or linear. This follows from an extended classification of preservers of these two properties for matrices of small size. We also prove that the same assertions hold upon working only with symmetric matrices; for total-positivity preservers our proofs proceed through solving two totally positive completion problems.

For each of several S G R", ", those linear transformations 9: R", "-+ R", n which map S onto S are characterized. Each class is a familiar one which generalizes the notion of positivity to matrices. The classes include: the matrices with nonnegative principal minors, the M-matrices, the totally nonnegative matrices, the Dstable

Linear Algebra and its Applications, 2006

The following theorem is proved. Theorem. Suppose M = (a i,j) be a k × k matrix with positive entries and a i,j a i+1,j+1 > 4 cos 2 π k+1 a i,j+1 a i+1,j (1 ≤ i ≤ k − 1, 1 ≤ j ≤ k − 1). Then det M > 0. The constant 4 cos 2 π k+1 in this Theorem is sharp. A few other results concerning totally positive and multiply positive matrices are obtained.

Journal of Mathematical Physics, 2009

The structure of cones of positive and k-positive maps acting on a finite-dimensional Hilbert space is investigated. Special emphasis is given to their duality relations to the sets of superpositive and k-superpositive maps. We characterize k-positive and k-superpositive maps with regard to their properties under taking compositions. A number of results obtained for maps are also rephrased for the corresponding cones of block positive, k-block positive, separable and k-separable operators, due to the Jamio lkowski-Choi isomorphism. Generalizations to a situation where no such simple isomorphism is available are also made, employing the idea of mapping cones. As a side result to our discussion, we show that extreme entanglement witnesses, which are optimal, should be of special interest in entanglement studies.

Linear and Multilinear Algebra, 1998

We establish a sufficient condition for strict total positivity of a matrix. In particular, we show that if the (positive) elements of a square matrix grow sufficiently fast as their distance from the diagonal of the matrix increases, then the matrix is strictly totally positive.

Linear Algebra and its Applications, 1976

Let K be a cone in R", K* its dual cone. An n X n matrix A is called cross-positive on K if and only if for all y E K, .z E K* such that (z, y) = 0 we have (z,Ay) > 0. In this short note new equivalent conditions for matrices cross-positive on K will be given in terms of the partial ordering in R" induced by the cone K.

Studia Mathematica, 2017

In a nutshell, we intend to extend Schoenberg's classical theorem connecting conditionally positive semidefinite functions F : R n → C, n ∈ N, and their positive semidefinite exponentials exp(tF ), t > 0, to the case of matrix-valued functions F : R n → C m×m , m ∈ N. Moreover, we study the closely associated property that exp(tF (-i∇)), t > 0, is positivity preserving and its failure to extend directly in the matrix-valued context.

Linear Algebra and its Applications, 1995

Some shape-preserving properties of positive linear operators, involving higher order convexity and Lipschitz classes, are investigated from the point of view of weak Tchebycheff systems and total positivity in the sense of Karlin [8]. The same properties are shown to be fulfilled by the strongly continuous semigroup (T (t)) t≥0 , if any, generated by the iterates of the relevant operators, in the spirit of Altomare's theory. MSC 2000. 41A36, 47D06.