Issue 4, Volume 43, December 2018

2018

1. Norm Ratios Under a Weak Order Relation in $\mathbb{M}_{m}\otimes \mathbb{M}_{n}$ Tsuyoshi Ando PDF

   ▸ Abstract

   In the real Hilbert space of self-adjoint elements of the tensor product ${\mathbb {M}}_{m}\otimes {\mathbb {M}}_{n}$, there are two natural cones besides the cone ${\mathfrak {P}}_{0}$ of positive semi-definite elements. The one is and the other is the cone ${\mathfrak {P}}_{-}$, dual to ${\mathfrak {P}}_{+}$ with respect to the inner product. Then, ${\mathfrak {P}}_{+} \subset {\mathfrak {P}}_{0} \subset {\mathfrak {P}}_{-}.$ A weak order relation $\geq $ is introduced by Our interest is in finding bounds for the ratio $|||T|||/|||S|||$ for $S \geq T \geq 0$, where $| | |\cdot| | |$ is one of the operator norm, the trace norm, and the Hilbert-Schmidt norm. $${\mathfrak{P}}_{+} := \text{the convex hull of}\ \{X\otimes Y ; ~ 0 \leq X \in {\mathbb{M}}_{m},~ 0 \leq Y \in {\mathbb{M}}_{n}\}$$ $$\mathbf{S} \succeq \mathbf{T}\quad \overset{def}{\Longleftrightarrow}\quad \mathbf{S} - \mathbf{T} \in {\mathfrak{P}}_{-}.$$

   585-594
2. A New Type of Operator Convexity Trung-Hoa Dinh , Thanh-Duc Dinh , Bich-Khue T. Vo PDF

   ▸ Abstract

   Let $r,s$ be positive numbers. We define a new class of operator $(r,s)$-convex functions by the following inequality $$f \left( \left[\lambda A^{r} + (1-\lambda)B^{r}\right]^{1/r}\right) \leq \left[\lambda f(A)^{s} +(1-\lambda)f(B)^{s}\right]^{1/s},$$ where $A,B$ are positive definite matrices and for any $\lambda\in[0,1]$. We prove the Jensen, Hansen-Pedersen, and Rado type inequalities for such functions. Some equivalent conditions for a function $f$ to become operator $(r,s)$-convex are established.

   595-605
3. Inequalities for Relative Operator Entropies and Operator Means Shigeru Furuichi , Nicuşor Minculete PDF

   ▸ Abstract

   The main purpose of this article is to study estimates for the Tsallis relative operator entropy, by using the Hermite-Hadamard inequality. We obtain alternative bounds for the Tsallis relative operator entropy and in the process to derive these bounds, we established the significant relation between the Tsallis relative operator entropy and the generalized relative operator entropy. In addition, we study the properties on monotonicity for the weight of operator means and for the parameter of relative operator entropies.

   607-618
4. Indecomposable Exposed Positive Bi-linear Maps Between Two by Two Matrices Seung-Hyeok Kye PDF

   ▸ Abstract

   Positive bi-linear maps between matrix algebras play important roles to detect tri-partite entanglement by the duality between bi-linear maps and tri-tensor products. We exhibit indecomposable positive bi-linear maps between $2 \times 2$ matrices which generate extreme rays in the cone of all positive bi-linear maps. In fact, they are exposed, and so detect entanglement of positive partial transpose (PPT) whose volume is nonzero.

   619-627
5. Completely Positive Matrices: Real, Rational, and Integral Abraham Berman , Naomi Shaked-Monderer PDF

   ▸ Abstract

   This paper was presented as an invited talk in the 6th International Conference on Matrix Analysis and Applications, Duy Tan University, Da Nang City, Vietnam, June 15–18, 2017. All the matrices in the paper are real. We survey known results and present some new problems. The paper has six parts.

   1. What is complete positivity?

   2. Why are completely positive matrices important?

   3. How can we tell if a given matrix is completely positive?

   4. Does every rational completely positive matrix have a rational cp-factorization?

   5. cp-rank.

   6. Which integral completely positive matrices have an integral cp-factorization?

   629-639
6. The Change in Multiplicity of an Eigenvalue of a Real Symmetric Matrix Resulting from the Changes in Edge Values Around a Classified Vertex in a Tree Kenji Toyonaga PDF

   ▸ Abstract

   Given a real symmetric matrix $A$ whose graph is a tree $T$ and its eigenvalues, vertices in $T$ can be classified in three categories, based upon the change in multiplicity of a particular eigenvalue, when the vertex is removed. We investigate the change in multiplicity of an eigenvalue based upon changes in edge values around a classified vertex. Then, we observe that a 2-Parter edge, a Parter edge, and a downer edge are located separately from each other in a tree, and there is a neutral edge between them. Especially, we show that the distance between a downer edge and a 2-Parter edge or a Parter edge is at least 2. We also consider some relations between a downer edge and the value on a downer vertex.

   641-647
7. Graphs with Three Distinct $\alpha$-Eigenvalues Muhammad Ateeq Tahir , Xiao-Dong Zhang PDF

   ▸ Abstract

   In this paper, we investigate properties of graphs with three distinct $\alpha$-eigenvalues of the matrix $A_{\alpha}$. In particular, we show for some $\alpha$ the connected graph $G = K_n − e,$ $G = K_1 \vee (nK_n)$ and some cones over strongly regular graph admit three distinct $\alpha$-eigenvalues.

   649-659
8. Generalizing Choi-Like Maps Dariusz Chruściński , Marcin Marciniak , Adam Rutkowski PDF

   ▸ Abstract

   A problem of further generalization of generalized Choi maps $\Phi_{[a,b,c]}$ acting on $\mathbb M_3$ introduced by Cho, Kye, and Lee is discussed. Some necessary conditions for positivity of the generalized maps are provided as well as some sufficient conditions. Also, some sufficient conditions for decomposability of these maps are shown.

   661-674
9. Symmetric, Hankel-Symmetric, and Centrosymmetric Doubly Stochastic Matrices Richard A. Brualdi , Lei Cao PDF

   ▸ Abstract

   We investigate convex polytopes of doubly stochastic matrices having special structures: symmetric, Hankel-symmetric, centrosymmetric, and both symmetric and Hankel-symmetric. We determine dimensions of these polytopes and classify their extreme points. We also determine a basis of the real vector spaces generated by permutation matrices with these special structures.

   675-700
10. The Permanent Functions of Tensors Qing-Wen Wang , Fuzhen Zhang PDF

    ▸ Abstract

    By a tensor we mean a multidimensional array (matrix) or hypermatrix over a number field. This article aims to set an account of the studies on the permanent functions of tensors. We formulate the definitions of 1-permanent, 2-permanent, and $k$-permanent of a tensor in terms of hyperplanes, planes, and $k$-planes of the tensor; we discuss the polytopes of stochastic tensors; at the end, we present an extension of the generalized matrix function for tensors.

    701-713
11. Spectral Characterizations of Solvability and Stability for Delay Differential-Algebraic Equations Phi Ha PDF

    ▸ Abstract

    The solvability and stability analyses of linear time invariant systems of delay differential-algebraic equations (DDAEs) are analyzed. The behavior approach is applied to DDAEs in order to establish characterizations of their solvability in terms of spectral conditions. Furthermore, examples are delivered to demonstrate that the eigenvalue-based approach in analyzing the exponential stability of dynamical systems is only valid for a special class of DDAEs, namely, non-advanced. Then, a new concept of weak stability is proposed and studied for DDAEs whose matrix coefficients pairwise commute.

    715-735
12. Parameterization of Translation-Invariant Two-Dimensional Two-State Quantum Walks Hiromichi Ohno PDF

    ▸ Abstract

    This study investigates unitary equivalence classes of translation-invariant two-dimensional two-state quantum walks. We show that unitary equivalence classes of such quantum walks are essentially parameterized by two real parameters.

    737-747
13. Rhaly Operators on Small Weighted Hardy Spaces Tan Pin Lin , Minh Luan Doan , Le Hai Khoi PDF

    ▸ Abstract

    We study several properties of Rhaly operators on small weighted Hardy spaces of holomorphic functions in the unit disc $\mathbb D$. In particular, we obtain criteria for boundedness, compactness, Hilbert-Schmidt norm, and $p$-Schatten class membership of such operators. A closedness of Rhaly operator is also studied.

    739-759
14. A Note on Nondegenerate Matrix Polynomials Trung Hoa Dinh , Toan Minh Ho , Tiến Sơn Phạm PDF

    ▸ Abstract

    n this paper, via Newton polyhedra, we define and study symmetric matrix polynomials which are nondegenerate at infinity. From this, we construct a class of (not necessarily compact) semialgebraic sets in $\mathbb R^n$ such that for each set $K$ in the class, we have the following two statements: (i) the space of symmetric matrix polynomials, whose eigenvalues are bounded on $K$, is described in terms of the Newton polyhedron corresponding to the generators of $K$ (i.e., the matrix polynomials used to define $K$) and is generated by a finite set of matrix monomials; and (ii) a matrix version of Schmüdgen’s Positivstellensätz holds: every matrix polynomial, whose eigenvalues are “strictly” positive and bounded on $K$, is contained in the preordering generated by the generators of $K$.

    761-778