On linear preservers of semipositive matrices

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.

On Properties of Semipositive Cones and 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.

Piecewise Linear Sheaves

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.

Semipositivity of linear maps relative to proper cones in finite dimensional real Hilbert spaces

For a proper cone $K$ in a finite dimensional real Hilbert space $V$, a linear map $L$ is said to be $K$-semipositive if there exists $d \in K^\circ$, the interior of $K$, such that $L(d) \in K^\circ$. The aim of this manuscript is to characterize $K$-semipositivity of linear maps relative to a proper cone. Among several results obtained, $K$-semipositivity is characterized in terms of products of the form $YX^{-1}$ for $K$-positive linear maps ($L(K \setminus \{0\}) \subseteq K^\circ$) with $X$ invertible, semipositivity of matrices relative to the $n$-dimensional Lorentz cone $\mathcal{L}^n_{+}$ is characterized, semipositivity of the following three linear maps relative to the cone $\mathcal{S}^n_{+}$: $X \mapsto AXB$ (denoted by $M_{A,B}$), $X \mapsto AXB + B^tXA^t$ (denoted by $L_{A,B}$), where $A, B \in M_n(\reals)$, and $X \mapsto X - AXA^t$ (denoted by $S_A$, known as the Stein transformation) is characterized. It is also proved that $M_{A,B}$ is semipositive if and only if $B = \alpha A^t$ for some $\alpha > 0$, the map $L_{A,B}$ is semipositive if and only if $A(B^t)^{-1}$ is positive stable. A particular case of the new result generalizes Lyapunov's theorem. Decompositions of the above maps (when they are semipositive) in the form $L_1L_2^{-1}$, where $L_1$ and $L_2$ are both positive and invertible (assuming $A$ is invertible in the case of $S_A$) are presented. Moreover, a question on invariance of the semipositive cone $\mathcal{K}_A$ of a matrix under $A$ is partially answered.

Eventual 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.

A SPECTRAL CHARACTERIZATION OF OPERATORS

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.