On the bilinearity rank of a proper cone and Lyapunov-like transformations

2013 ◽  
Vol 147 (1-2) ◽  
pp. 155-170 ◽  
Author(s):  
M. Seetharama Gowda ◽  
Jiyuan Tao
Keyword(s):  
Proper Cone

scholarly journals A SPECTRAL CHARACTERIZATION OF OPERATORS

2016 ◽  
Vol 102 (3) ◽  
pp. 369-391 ◽  
Author(s):  
SATISH K. PANDEY ◽  
VERN I. PAULSEN
Keyword(s):  
Banach Space ◽  
Hilbert Spaces ◽  
Real Banach Space ◽  
Closed Subspace ◽  
Characterization Theorem ◽  
Positive Operators ◽  
Spectral Characterization ◽  
Proper Cone ◽  
Arbitrary Dimensions

We establish a spectral characterization theorem for the operators on complex Hilbert spaces of arbitrary dimensions that attain their norm on every closed subspace. The class of these operators is not closed under addition. Nevertheless, we prove that the intersection of these operators with the positive operators forms a proper cone in the real Banach space of hermitian operators.

An improved bound for the Lyapunov rank of a proper cone

2015 ◽  
Vol 10 (1) ◽  
pp. 11-17 ◽  
Author(s):  
Michael Orlitzky ◽  
M. Seetharama Gowda
Keyword(s):  
Proper Cone ◽  
Lyapunov Rank

scholarly journals Sur les mesures coniques localisables

1982 ◽  
Vol 33 (3) ◽  
pp. 394-400 ◽  
Author(s):  
Richard Becker
Keyword(s):  
Linear Form ◽  
Convex Space ◽  
Radon Measure ◽  
Locally Convex Space ◽  
Proper Cone ◽  
Locally Convex

AbstractLet X be a weakly complete proper cone, contained in an Hausdorff locally convex space E, with continuous dual E′. A positive linear form on the Risz space of functions on X generated by E′ is called a conical measure on X. Let M+ (X) be the set of all conical measures on X. G. Choquet asked the question: when is every conical measure on X given by a Radon measure on (X\0)? Let L be the class of such X. In this paper we show that the fact that X ∈ L only depends, in some sense, on the cofinal subsets of the space E′|x ordered by the order of functions on X. We derive that X ∈ L is equivalent to M+ (M) ∈ L. We show that is closed under denumerable products.

scholarly journals On linear preservers of semipositive matrices

2021 ◽  
Vol 37 (37) ◽  
pp. 88-112
Author(s):  
Sachindranath Jayaraman ◽  
Vatsalkumar Mer
Keyword(s):  
Left Inverse ◽  
Linear Preserver ◽  
Proper Cone ◽  
Linear Preservers ◽  
Linear Preserver Problems ◽  
Preserver Problems

Given proper cones $K_1$ and $K_2$ in $\mathbb{R}^n$ and $\mathbb{R}^m$, respectively, an $m \times n$ matrix $A$ with real entries is said to be semipositive if there exists a $x \in K_1^{\circ}$ such that $Ax \in K_2^{\circ}$, where $K^{\circ}$ denotes the interior of a proper cone $K$. This set is denoted by $S(K_1,K_2)$. We resolve a recent conjecture on the structure of into linear preservers of $S(\mathbb{R}^n_+,\mathbb{R}^m_+)$. We also determine linear preservers of the set $S(K_1,K_2)$ for arbitrary proper cones $K_1$ and $K_2$. Preservers of the subclass of those elements of $S(K_1,K_2)$ with a $(K_2,K_1)$-nonnegative left inverse as well as connections between strong linear preservers of $S(K_1,K_2)$ with other linear preserver problems are considered.

Eventual Cone Invariance

2017 ◽  
Vol 32 ◽  
pp. 204-216 ◽  
Author(s):  
Michael Kasigwa ◽  
Michael Tsatsomeros
Keyword(s):  
Dynamical System ◽  
Matrix Exponential ◽  
Nonnegative Matrices ◽  
Proper Cone ◽  
The Matrix ◽  
Cone Invariance

Eventually nonnegative matrices are square matrices whose powers become and remain (entrywise) nonnegative. Using classical Perron-Frobenius theory for cone preserving maps, this notion is generalized to matrices whose powers eventually leave a proper cone K ⊂ R^n invariant, that is, A^mK ⊆ K for all sufficiently large m. Also studied are the related notions of eventual cone invariance by the matrix exponential, as well as other generalizations of M-matrix and dynamical system notions.

scholarly journals Piecewise Linear Sheaves

2019 ◽  
Author(s):  
Masaki Kashiwara ◽  
Pierre Schapira
Keyword(s):  
Vector Space ◽  
Piecewise Linear ◽  
Building Blocks ◽  
Convex Polyhedra ◽  
Triangulated Category ◽  
Real Vector ◽  
Proper Cone ◽  
Finite Dimensional ◽  
Higher Dimensional

Abstract On a finite-dimensional real vector space, we give a microlocal characterization of (derived) piecewise linear sheaves (PL sheaves) and prove that the triangulated category of such sheaves is generated by sheaves associated with convex polyhedra. We then give a similar theorem for PL $\gamma $-sheaves, that is, PL sheaves associated with the $\gamma $-topology, for a closed convex polyhedral proper cone $\gamma $. Our motivation is that convex polyhedra may be considered as building blocks for higher dimensional barcodes.

scholarly journals On Properties of Semipositive Cones and Simplicial Cones

2020 ◽  
Vol 36 (36) ◽  
pp. 764-772
Author(s):  
Aritra Narayan Hisabia ◽  
Manideepa Saha
Keyword(s):  
Polyhedral Cone ◽  
Invariant Cone ◽  
Simplicial Cone ◽  
Positive Orthant ◽  
Proper Cone ◽  
Invertible Matrices ◽  
Simplicial Cones

For a given nonsingular $n\times n$ matrix $A$, the cone $S_{A}=\{x:Ax\geq 0\}$ , and its subcone $K_A$ lying on the positive orthant, called as semipositive cone, are considered. If the interior of the semipositive cone $K_A$ is not empty, then $A$ is named as semipositive matrix. It is known that $K_A$ is a proper polyhedral cone. In this paper, it is proved that $S_{A}$ is a simplicial cone and properties of its extremals are analyzed. An one-one relation between simplicial cones and invertible matrices is established. For a proper cone $K$ in $\mathbb{R}^n$, $\pi(K)$ denotes the collection of $n\times n$ matrices that leave $K$ invariant. For a given minimally semipositive matrix (no column-deleted submatrix is semipositive) $A$, it is shown that the invariant cone $\pi(K_A)$ is a simplicial cone.

scholarly journals Mesures coniques et intégrale de Daniell

1981 ◽  
Vol 31 (1) ◽  
pp. 46-58 ◽  
Author(s):  
Richard Becker
Keyword(s):  
Linear Form ◽  
Riesz Space ◽  
Integration Theory ◽  
Continuous Linear ◽  
Linear Forms ◽  
Proper Cone ◽  
Weak Space

AbstractLet X be a weakly complete proper cone contained in a weak space E and h(E) the Riesz space generated by the continuous linear forms on E. A positive conical measure μ on X is a positive linear form on h(E)|x. G. Choquet has proved μ is a Daniell integral on E when E is weakly complete, but μ is not generally a Daniell integral on X. However we give an integration theory for functions on X and compare this theory with the classical Daniell theory. The case where μ is maximal in the sense of G. Choquet is remarkable.

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