arXiv subjects

M. S. Moslehian

Publications and source records attributed to M. S. Moslehian.

At least 37 records · Page 2Linked to original sources

arXiv2018

An extension of orthogonality relations based on norm derivatives

We introduce the relation $ρ_λ$-orthogonality in the setting of normed spaces as an extension of some orthogonality relations based on norm derivatives, and present some of its essential properties. Among other things, we give a characterization of inner product spaces via the functional $ρ_λ$. Moreover, we consider a class of linear mappings preserving this new kind of orthogonality. In particular, we show that a linear mapping preserving $ρ_λ$-orthogonality has to be a similarity, that is, a scalar multiple of an isometry.

math.FA↗

arXiv2017

Orthogonality and parallelism of operators on various Banach spaces

We present some properties of orthogonality and relate them with support disjoint and norm inequalities in p Schatten ideals. In addition, we investigate the problem of characterization of norm parallelism for bounded linear operators. We consider the characterization of norm parallelism problem in p Schatten ideals and locally uniformly convex spaces. Later on, we study the case when an operator is norm parallel to the identity operator. Finally, we give some equivalence assertions about the norm parallelism of compact operators. Some applications and generalizations are discussed for certain operators.

math.FA↗

arXiv2017

Norm inequalities related to the Heron and Heinz means

In this article, we present several inequalities treating operator means and the Cauchy-Schwarz inequality. In particular, we present some new comparisons between operator Heron and Heinz means, several generalizations of the difference version of the Heinz means and further refinements of the Cauchy-Schwarz inequality. The techniques used to accomplish these results include convexity and Löwner matrices.

math.FA↗

arXiv2017

Matrix KSGNS construction and a Radon--Nikodym type theorem

In this paper, we introduce the concept of completely positive matrix of linear maps on Hilbert $A$-modules over locally $C^{*}$-algebras and prove an analogue of Stinespring theorem for it. We show that any two minimal Stinespring representations for such matrices are unitarily equivalent. Finally, we prove an analogue of the Radon--Nikodym theorem for this type of completely positive $n\times n$ matrices.

math.OA↗

arXiv2017

Relative Entropy and Tsallis Entropy of two Accretive Operators

Let $A$ and $B$ be two accretive operators. We first introduce the weighted geometric mean of $A$ and $B$ together with some related properties. Afterwards, we define the relative entropy as well as the Tsallis entropy of $A$ and $B$. The present definitions and their related results extend those already introduced in the literature for positive invertible operators.

math.FA↗

arXiv2017

Geometric aspects of $p$-angular and skew $p$-angular distances

Corresponding to the concept of $p$-angular distance $α_p[x,y]:=\left\lVert\lVert x\rVert^{p-1}x-\lVert y\rVert^{p-1}y\right\rVert$, we first introduce the notion of skew $p$-angular distance $β_p[x,y]:=\left\lVert \lVert y\rVert^{p-1}x-\lVert x\rVert^{p-1}y\right\rVert$ for non-zero elements of $x, y$ in a real normed linear space and study some of significant geometric properties of the $p$-angular and the skew $p$-angular distances. We then give some results comparing two different $p$-angular distances with each other. Finally, we present some characterizations of inner product spaces related to the $p$-angular and the skew $p$-angular distances. In particular, we show that if $p>1$ is a real number, then a real normed space $\mathcal{X}$ is an inner product space, if and only if for any $x,y\in \mathcal{X}\smallsetminus{\lbrace 0\rbrace}$, it holds that $α_p[x,y]\geqβ_p[x,y]$.

math.FA↗

arXiv2016

Operator equations $AX+YB=C$ and $AXA^+BYB^=C$ in Hilbert $C^*$-modules

Let $A,B$ and $C$ be adjointable operators on a Hilbert $C^*$-module $\mathscr{E}$. Giving a suitable version of the celebrated Douglas theorem in the context of Hilbert $C^*$-modules, we present the general solution of the equation $AX+YB=C$ when the ranges of $A,B$ and $C$ are not necessarily closed. We examine a result of Fillmore and Williams in the setting of Hilbert $C^*$-modules. Moreover, we obtain some necessary and sufficient conditions for existence of a solution for $AXA^*+BYB^*=C$. Finally, we deduce that there exist nonzero operators $X, Y\geq 0$ and $Z$ such that $AXA^*+BYB^*=CZ$, when $A, B$ and $C$ are given subject to some conditions.

math.OA↗

arXiv2016

Quantum information inequalities via tracial positive linear maps

We present some generalizations of quantum information inequalities involving tracial positive linear maps between $C^*$-algebras. Among several results, we establish a noncommutative Heisenberg uncertainty relation. More precisely, we show that if $Φ: \mathcal{A} \to \mathcal{B}$ is a tracial positive linear map between $C^*$-algebras , $ρ\in \mathcal{A}$ is a $Φ$-density element and $A,B$ are self-adjoint operators of $\mathcal{A}$ such that $ {\rm sp}(\mbox{-i}ρ^\frac{1}{2}[A,B]ρ^\frac{1}{2}) \subseteq [m,M] $ for some scalers $0<m<M$, then under some conditions \begin{eqnarray}\label{inemain1} V_{ρ,Φ}(A)\sharp V_{ρ,Φ}(B)\geq \frac{1}{2\sqrt{K_{m,M}(ρ[A,B])}} \left|Φ(ρ[A,B])\right|, \end{eqnarray} where $K_{m,M}(ρ[A,B])$ is the Kantorovich constant of the operator $\mbox{-i}ρ^\frac{1}{2}[A,B]ρ^\frac{1}{2}$ and $V_{ρ,Φ}(X)$ is the generalized variance of $X$.\\ In addition, we use some arguments differing from the scalar theory to present some inequalities related to the generalized correlation and the generalized Wigner--Yanase--Dyson skew information.

math.FA↗

arXiv2016

An extension of the Pólya--Szegö operator inequality

We extend an operator Pólya--Szegö type inequality involving the operator geometric mean to any arbitrary operator mean under some mild conditions. Utilizing the Mond--Pečarić method, we present some other related operator inequalities as well.

math.FA↗

arXiv2016

Estimates of operator convex and operator monotone functions on bounded intervals

Recently the behavior of operator monotone functions on unbounded intervals with respect to the relation of strictly positivity has been investigated. In this paper we deeply study such behavior not only for operator monotone functions but also for operator convex functions on bounded intervals. More precisely, we prove that if $f$ is a nonlinear operator convex function on a bounded interval $(a,b)$ and $A, B$ are bounded linear operators acting on a Hilbert space with spectra in $(a,b)$ and $A-B$ is invertible, then $sf(A)+(1-s)f(B)>f(sA+(1-s)B)$. A short proof for a similar known result concerning a nonconstant operator monotone function on $[0,\infty)$ is presented. Another purpose is to find a lower bound for $f(A)-f(B)$, where $f$ is a nonconstant operator monotone function, by using a key lemma. We also give an estimation of the Furuta inequality, which is an excellent extension of the Löwner--Heinz inequality.

math.FA↗

arXiv2016

A New Class of Operator Monotone Functions via Operator Means

In this paper, we obtain a new class of functions, which is developed via the Hermite--Hadamard inequality for convex functions. The well-known one-one correspondence between the class of operator monotone functions and operator connections declares that the obtained class represents the weighted logarithmic means. We shall also consider weighted identric mean and some relationships between various operator means. Among many things, we extended the weighted arithmetic--geometric operator mean inequality as $A\#_{t}B\leq A\ell_t B\leq \frac{1}{2}(A\#_{t}B + A\nabla_{t} B)\le A\nabla_tB$ and $A\#_{t}B\leq A\mathcal{I}_t B\leq A\nabla_{t} B$ involving the considered operator means.

math.FA↗

arXiv2015

Extension of Euclidean operator radius inequalities

To extend the Euclidean operator radius, we define $w_p$ for an $n$-tuples of operators $(T_1,\ldots, T_n)$ in $\mathbb{B}(\mathscr{H})$ by $w_p(T_1,\ldots,T_n):= \sup_{\| x \| =1} \left(\sum_{i=1}^{n}| \langle T_i x, x \rangle |^p \right)^{\frac1p}$ for $p\geq1$. We generalize some inequalities including Euclidean operator radius of two operators to those involving $w_p$. Further we obtain some lower and upper bounds for $w_p$. Our main result states that if $f$ and $g$ are nonnegative continuous functions on $\left[ 0,\infty \right) $ satisfying $f\left( t\right) g\left(t\right) =t$ for all $t\in \left[ 0,\infty \right) $, then \begin{equation*} w_{p}^{rp}\left( A_{1}^{\ast }T_{1}B_{1},\ldots ,A_{n}^{\ast }T_{n}B_{n}\right) \leq \frac{1}{2}\left\Vert \underset{i=1}{\overset{n}{\sum }}\Big( \left[ B_{i}^{\ast }f^{2}\left( \left\vert T_{i}\right\vert \right) B_{i}\right] ^{rp}+\left[ A_{i}^{\ast }g^{2}\left( \left\vert T_{i}^{\ast }\right\vert \right) A_{i}\right] ^{rp}\Big)\right\Vert \end{equation*} for all $p\geq 1$, $r\geq 1$ and operators in $ \mathbb{B}(\mathscr{H})$.

math.FA↗

arXiv2014

Complementary and refined inequalities of Callebaut inequality for operators

The Callebaut inequality says that \begin{align*} \sum_{ j=1}^n \left(A_j\sharp B_j\right)\leq \left(\sum_{ j=1}^n A_j σB_j\right)\sharp\left(\sum_{ j=1}^n A_j σ^{\bot} B_j\right)\leq\left(\sum_{ j=1}^n A_j\right)\sharp \left(\sum_{ j=1}^nB_j\right)\,, \end{align*} where $A_j, B_j\,\,(1\leq j\leq n)$ are positive invertible operators and $σ$ and $σ^\perp$ are an operator mean and its dual in the sense of Kabo and Ando, respectively. In this paper we employ the Mond--Pečarić method as well as some operator techniques to establish a complementary inequality to the above one under mild conditions. We also present some refinements of a Callebaut type inequality involving the weighted geometric mean and Hadamard products of Hilbert space operators.

math.FA↗

arXiv2014

Inequalities for trace on $τ$-measurable operators

Let $\mathfrak{M}$ be a semifinite von Neumann algebra on a Hilbert space equipped with a faithful normal semifinite trace $τ$. A closed densely defined operator $x$ affiliated with $\mathfrak{M}$ is called $τ$-measurable if there exists a number $λ\geq 0$ such that $τ\left(e^{|x|}(λ,\infty)\right)<\infty$. A number of useful inequalities, which are known for the trace on Hilbert space operators, are extended to trace on $τ$-measurable operators. In particular, these inequalities imply Clarkson inequalities for $n$-tuples of $τ$-measurable operators. A general parallelogram law for $τ$-measurable operators are given as well.

math.OA↗

arXiv2014

Operator convexity in Krein spaces

We introduce the notion of Krein-operator convexity in the setting of Krein spaces. We present an indefinite version of the Jensen operator inequality on Krein spaces by showing that if $(\mathscr{H},J)$ is a Krein space, $\mathcal{U}$ is an open set which is symmetric with respect to the real axis such that $\mathcal{U}\cap\mathbb{R}$ consists of a segment of real axis and $f$ is a Krein-operator convex function on $\mathcal{U}$ with $f(0)=0$, then \begin{eqnarray*} f(C^{\sharp}AC)\leq^{J}C^{\sharp}f(A)C \end{eqnarray*} for all $J$-positive operators $A$ and all invertible $J$-contractions $C$ such that the spectra of $A$, $C^{\sharp}AC$ and $D^{\sharp}AD$ are contained in $\mathcal{U}$, where $D$ is a defect operator for $C^{\sharp}$.\\ We also show that in contrast with usual operator convex functions the converse of this implication is not true, in general.

math.FA↗

arXiv2013

Deformation of involution and multiplication in a C*-algebra

We investigate the deformation of involution and multiplication in a unital $C^*$-algebra when its norm is fixed. Our main result is to present all multiplications and involutions on a given $C^*$-algebra $\mathcal{A}$ under which $\mathcal{A}$ is still a $C^*$-algebra whereas we keep the norm unchanged. For each invertible element $a\in\mathcal{A}$ we also introduce an involution and a multiplication making $\mathcal{A}$ into a $C^*$-algebra in which $a$ becomes a positive element. Further, we give a necessary and sufficient condition for that the center of a unital $C^*$-algebra $\mathcal{A}$ is trivial.

math.OA↗

arXiv2013

Bellman inequality for Hilbert space operators

We establish some operator versions of Bellman's inequality. In particular, we prove that if $Φ: \mathbb{B}(\mathscr{H}) \to \mathbb{B}(\mathscr{K})$ is a unital positive linear map, $A,B \in \mathbb{B}(\mathscr{H})$ are contractions, $p>1$ and $0 \leq λ\leq 1$, then {eqnarray*} \big(Φ(I_\mathscr{H}-A\nabla_λB)\big)^{1/p}\geΦ\big((I_\mathscr{H}-A)^{1/p}\nabla_λ(I_\mathscr{H}-B)^{1/p}\big). {eqnarray*}

math.FA↗

arXiv2013

Operator inequalities of Jensen type

We present some generalized Jensen type operator inequalities involving sequences of self-adjoint operators. Among other things, we prove that if $f:[0,\infty) \to \mathbb{R}$ is a continuous convex function with $f(0)\leq 0$, then {equation*} \sum_{i=1}^{n} f(C_i) \leq f(\sum_{i=1}^{n}C_i)-δ_f\sum_{i=1}^{n}\widetilde{C}_i\leq f(\sum_{i=1}^{n}C_i) {equation*} for all operators $C_i$ such that $0 \leq C_i\leq M \leq \sum_{i=1}^{n} C_i $ \ $(i=1,...,n)$ for some scalar $M\geq0$, where $ \widetilde{C_i} = 1/2 - |\frac{C_i}{M}- 1/2 |$ and $δ_f = f(0)+f(M) - 2 f(\frac{M}{2})$.

math.FA↗