Related papers: Linear mappings preserving the copositive cone

Let $A_1, \ldots, A_r$ be Hermitian $n \times n$ matrices and $S = \sum A_i \otimes s_i$ the associated matrix semicircle, where $s_1, \ldots, s_r$ are free semicircular variables. We prove that the following are equivalent: (i) the matrix…

A symmetric matrix $C$ is completely positive (CP) if there exists an entrywise nonnegative matrix $B$ such that $C=BB^T$. The CP-completion problem is to study whether we can assign values to the missing entries of a partial matrix (i.e.,…

The dual of a matrix ordered space has a natural matrix ordering that makes the dual space matrix ordered as well. The purpose of these notes is to give a condition that describes when the linear map taking a basis of the n by n matrices to…

We investigate the completely positive semidefinite cone $\mathcal{CS}_+^n$, a new matrix cone consisting of all $n\times n$ matrices that admit a Gram representation by positive semidefinite matrices (of any size). In particular we study…

We first show that a continuous function f is nonnegative on a closed set $K\subseteq R^n$ if and only if (countably many) moment matrices of some signed measure $d\nu =fd\mu$ with support equal to K, are all positive semidefinite (if $K$…

We prove that the fixed point iteration of arbitrary positive concave mappings with nonempty fixed point set converges geometrically for any starting point. We also show that positivity is crucial for this result to hold, and the concept of…

The work considers an equivalence relation in the set of all $n\times m$ matrices with entries in the set $[p]=\{ 0,1,\ldots , p-1 \}$. In each element of the factor-set generated by this relation, we define the concept of canonical matrix,…

We investigate the iterative behaviour of continuous order preserving subhomogeneous maps that map a polyhedral cone into itself. For these maps we show that every bounded orbit converges to a periodic orbit and, moreover, that there exists…

A map between operator spaces is called completely coarse if the sequence of its amplifications is equi-coarse. We prove that all completely coarse maps must be $\mathbb R$-linear. On the opposite direction of this result, we introduce a…

We consider maps defined on the interior of a normal, closed cone in a real Banach space that are nonexpansive with respect to Thompson's metric. With mild compactness assumptions, we prove that the Krasnoselskii iterates of such maps…

We present some properties of (not necessarily linear) positive maps between $C^*$-algebras. We first extend the notion of Lieb functions to that of Lieb positive maps between $C^*$-algebras. Then we give some basic properties and…

A recently-established necessary condition for polynomials that preserve the class of entrywise nonnegative matrices of a fixed order is shown to be necessary and sufficient for the class of nonnegative monomial matrices. Along the way, we…

It is proved that a map $\varphi\colon R\to S$ of commutative noetherian rings that is essentially of finite type and flat is locally complete intersection if and only $S$ is proxy small as a bimodule. This means that the thick subcategory…

In this paper we study the positive definiteness of meet and join matrices using a novel approach. When the set $S_n$ is meet closed, we give a sufficient and necessary condition for the positive definiteness of the matrix $(S_n)_f$. From…

We consider a large class of matrix problems, which includes the problem of classifying arbitrary systems of linear mappings. For every matrix problem from this class, we construct Belitskii's algorithm for reducing a matrix to a canonical…

Let $K$ be a field, $V$ a finite dimensional $K$-vector space and $E$ the exterior algebra of $V$. We analyze iterated mapping cone over $E$. If $I$ is a monomial ideal of $E$ with linear quotients, we show that the mapping cone…

Let $X\subset \mathbb{C}^n$ be a smooth irreducible affine variety of dimension $k$ and let $F: X\to \mathbb{C}^m$ be a polynomial mapping. We prove that if $m\ge k$, then there is a Zariski open dense subset $U$ in the space of linear…

This paper investigates the uniqueness of a nonnegative vector solution and the uniqueness of a positive semidefinite matrix solution to underdetermined linear systems. A vector solution is the unique solution to an underdetermined linear…

Let K be an arbitrary (commutative) field, and V be a linear subspace of M_n(K) such that codim V<n-1. Using a recent generalization of a theorem of Atkinson and Lloyd, we show that every linear embedding of V into M_n(K) which strongly…

The (matricial) solution set of a Linear Matrix Inequality (LMI) is a convex basic non-commutative semi-algebraic set. The main theorem of this paper is a converse, a result which has implications for both semidefinite programming and…