arXiv subjects

Mohammad Sal Moslehian

Publications and source records attributed to Mohammad Sal Moslehian.

At least 37 records · Page 2Linked to original sources

arXiv2019

Power means of probability measures and Ando-Hiai inequality

Let $μ$ be a probability measure of compact support on the set $\mathbb{P}_n$ of all positive definite matrices, let $t\in(0,1]$, and let $P_t(μ)$ be the unique positive solution of $X=\int_{\mathbb{P}_n}X\sharp_t Z dμ(Z)$. In this paper, we show that $$ P_t(μ)\leq I\quad \Longrightarrow\quad P_{\frac{t}{p}}(ν)\leq P_t(μ)$$ for every $p\geq1$, where $ν(Z)=μ(Z^{1/p})$. This provides an extension of the Ando--Hiai inequality for matrix power means. Moreover, we prove that if $Φ:\mathbb{M}_n\to\mathbb{M}_m$ is a unital positive linear map, then $Φ(P_t(μ))\leq P_t(ν)$ for all $t\in[-1,1]\backslash\{0\}$, where $ν$ is a certain measure.

math.FA↗

arXiv2019

Noncommutative versions of inequalities in quantum information theory

In this paper, we aim to replace in the definitions of covariance and correlation the usual trace {\rm Tr} by a tracial positive map between unital $C^*$-algebras and to replace the functions $x^α$ and $x^{1-α}$ by functions $f$ and $g$ satisfying some mild conditions. These allow us to define the generalized covariance, the generalized variance, the generalized correlation and the generalized Wigner--Yanase--Dyson skew information related to the tracial positive maps and functions $f$ and $g$. We persent a generalization of Heisenberg's uncertainty relation in the noncommutative framework. We extend some inequalities and properties for the generalized correlation and the generalized Wigner--Yanase--Dyson skew information. Furthermore, we extend some inequalities for the generalized skew information such as uncertainty relation and the relation between the generalized variance and the generalized skew information.

math.OA↗

arXiv2019

Halmos' two projections theorem for Hilbert $C^*$-module operators and the Friedrichs

Halmos' two projections theorem for Hilbert space operators is one of the fundamental results in operator theory. In this paper, we introduce the term of two harmonious projections in the context of adjointable operators on Hilbert $C^*$-modules, extend Halmos' two projections theorem to the case of two harmonious projections. We also give some new characterizations of the closed submodules and their associated projections. As an application, a norm equation associated to a characterization of the Friedrichs angle is proved to be true in the framework of Hilbert $C^*$-modules.

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arXiv2019

Positivity of $2\times 2$ block matrices of operators

We review some of the significant generalizations and applications of the celebrated Douglas theorem on the equivalence of factorization, range inclusion, and majorization of operators. We then apply it to find a characterization of the positivity of $2\times 2$ block matrices of operators in Hilbert spaces and finally describe the nature of such block matrices and provide several ways for showing their positivity.

math.FA↗

arXiv2018

Non-commutative Stein inequality and its applications

The non-commutative Stein inequality asks whether there exists a constant $C_{p,q}$ depending only on $p, q$ such that \begin{equation*} \left\| \left(\sum_{n} |\mathcal{E}_{n} (x_n) |^{q}\right)^{\frac{1}{q}} \right\|_p \leq C_{p,q} \left\| \left(\sum_{n} | x_n |^q \right)^{\frac{1}{q}}\right \|_p\qquad \qquad (S_{p,q}), \end{equation*} for (positive) sequences $(x_n)$ in $L_p(\mathcal{M})$. The validity of $(S_{p,2})$ for $1 < p < \infty$ and $(S_{p,1})$ for $1 \leq p < \infty$ are known. In this paper, we verify (i) $(S_{p,\infty})$ for $1 < p \leq \infty$; (ii) $(S_{p,p})$ for $1 \leq p < \infty$; (iii) $(S_{p,q})$ for $1 \leq q \leq 2$ and $q<p<\infty$. We also present some applications.

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arXiv2018

A variance bound for a general function of independent noncommutative random variables

The main purpose of this paper is to establish a noncommutative analogue of the Efron--Stein inequality, which bounds the variance of a general function of some independent random variables. Moreover, we state an operator version including random matrices, which extends a result of D. Paulin et al. [Ann. Probab. 44 (2016), no. 5, 3431--3473]. Further, we state a Steele type inequality in the framework of noncommutative probability spaces.

math.FA↗

arXiv2018

Approximately angle preserving mappings

In this paper, we present some characterizations of linear mappings, which preserve vectors at a specific angle. We introduce the concept of $(\varepsilon, c)$-angle preserving mappings for $|c|<1$ and $0\leq \varepsilon < 1 + |c|$. In addition, we define $\widehat{\varepsilon}\,(T, c)$ as the ``smallest'' number $\varepsilon$ for which $T$ is $(\varepsilon, c)$-angle preserving mapping. We state some properties of the function $\widehat{\varepsilon}\,(., c)$, and then propose an exact formula for $\widehat{\varepsilon}\,(T, c)$ in terms of the norm $\|T\|$ and the minimum modulus $[T]$ of $T$. Finally, we characterize the approximately angle preserving mappings.

math.FA↗

arXiv2017

On majorization and range inclusion of operators on Hilbert $C^*$-modules

It is proved that for adjointable operators $A$ and $B$ between Hilbert $C^*$-modules, certain majorization conditions are always equivalent without any assumptions on $\overline{\mathcal{R}(A^*)}$, where $A^*$ denotes the adjoint operator of $A$ and $\overline{\mathcal{R}(A^*)}$ is the norm closure of the range of $A^*$. In the case that $\overline{{\mathcal R}(A^*)}$ is not orthogonally complemented, it is proved that there always exists an adjointable operator $B$ whose range is contained in that of $A$, whereas the associated equation $AX=B$ for adjointable operators is unsolvable.

math.OA↗

arXiv2017

Operator maps of Jensen-type

Let $\mathbb{B}_J(\mathcal H)$ denote the set of self-adjoint operators acting on a Hilbert space $\mathcal{H}$ with spectra contained in an open interval $J$. A map $Φ\colon\mathbb{B}_J(\mathcal H)\to {\mathbb B}(\mathcal H)_\text{sa} $ is said to be of Jensen-type if \[ Φ(C^*AC+D^*BD)\le C^*Φ(A)C+D^*Φ(B)D \] for all $ A, B \in B_J(\mathcal H)$ and bounded linear operators $ C,D $ acting on $ \mathcal H $ with $ C^*C+D^*D=I$, where $I$ denotes the identity operator. We show that a Jensen-type map on a infinite dimensional Hilbert space is of the form $Φ(A)=f(A)$ for some operator convex function $ f $ defined in $ J $.

math.FA↗

arXiv2017

Quadratic interpolation of the Heinz means

The main goal of this article is to present several quadratic refinements and reverses of the well known Heinz inequality, for numbers and matrices, where the refining term is a quadratic function in the mean parameters. The proposed idea introduces a new approach to these inequalities, where polynomial interpolation of the Heinz function plays a major role. As a consequence, we obtain a new proof of the celebrated Heron-Heinz inequality proved by Bhatia, then we study an optimization problem to find the best possible refinement. As applications, we present matrix versions including unitarily invariant norms, trace and determinant versions.

math.FA↗

arXiv2017

Etemadi and Kolmogorov inequalities in noncommutative probability spaces

Based on a maximal inequality type result of Cuculescu, we establish some noncommutative maximal inequalities such as Hajék--Penyi inequality and Etemadi inequality. In addition, we present a noncommutative Kolmogorov type inequality by showing that if $x_1, x_2, \ldots, x_n$ are successively independent self-adjoint random variables in a noncommutative probability space $(\mathfrak{M}, τ)$ such that $τ\left(x_k\right) = 0$ and $s_k s_{k-1} = s_{k-1} s_k$, where $s_k = \sum_{j=1}^k x_j$, then for any $λ> 0$ there exists a projection $e$ such that $$1 - \frac{(λ+ \max_{1 \leq k \leq n} \|x_k\|)^2}{\sum_{k=1}^n {\rm var}(x_k)}\leq τ(e)\leq \frac{τ(s_n^2)}{λ^2}.$$ As a result, we investigate the relation between convergence of a series of independent random variables and the corresponding series of their variances.

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arXiv2017

Uffe Haagerup - His life and mathematics

In remembrance of Professor Uffe Valentin Haagerup (1949--2015), as a brilliant mathematician, we review some aspects of his life, and his outstanding mathematical accomplishments.

math.FA↗

arXiv2017

Characterizations of smooth spaces by $ρ_*$-orthogonality

The aim of this paper is to present some results concerning the $ρ_*$-orthogonality in real normed spaces and its preservation by linear operators. Among other things, we prove that if $T\,: X \longrightarrow Y$ is a nonzero linear $(I, ρ_*)$-orthogonality preserving mapping between real normed spaces, then $$\frac{1}{3}\|T\|\|x\|\leq\|Tx\|\leq 3[T]\|x\|, \qquad (x\in X)$$ where $[T]:=\inf\{\|Tx\|: \,x\in X, \|x\|=1\}$. We also show that the pair $(X,\perp_{ρ_*})$ is an orthogonality space in the sense of Rätz. Some characterizations of smooth spaces are given based on the $ρ_*$-orthogonality.

math.FA↗

arXiv2017

Noncommutative Blackwell-Ross martingale inequality

We establish a noncommutative Blackwell--Ross inequality for supermartingales under a suitable condition which generalize Khan's works to the noncommutative setting. We then employ it to deduce an Azuma-type inequality.

math.OA↗

arXiv2017

Solutions of the system of operator equations $BXA=B=AXB$ via $*$-order

In this paper, we establish some necessary and sufficient conditions for the existence of solutions to the system of operator equations $ BXA=B=AXB $ in the setting of bounded linear operators on a Hilbert space, where the unknown operator $X$ is called the inverse of $A$ along $B$. After that, under some mild conditions we prove that an operator $X$ is a solution of $ BXA=B=AXB $ if and only if $B \stackrel{*}{ \leq} AXA$, where the $*$-order $C\stackrel{*}{ \leq} D$ means $CC^*=DC^*, C^*C=C^*D$. Moreover we present the general solution of the equation above. Finally, we present some characterizations of $C \stackrel{*}{ \leq} D$ via other operator equations.

math.FA↗

arXiv2017

Characterizations of operator Birkhoff-James orthogonality

In this paper, we obtain some characterizations of the (strong) Birkhoff--James orthogonality for elements of Hilbert $C^*$-modules and certain elements of $\mathbb{B}(\mathscr{H})$. Moreover, we obtain a kind of Pythagorean relation for bounded linear operators. In addition, for $T\in \mathbb{B}(\mathscr{H})$ we prove that if the norm attaining set $\mathbb{M}_T$ is a unit sphere of some finite dimensional subspace $\mathscr{H}_0$ of $\mathscr{H}$ and $\|T\|_{{\mathscr{H}_0}^\perp} < \|T\|$, then for every $S\in\mathbb{B}(\mathscr{H})$, $T$ is the strong Birkhoff--James orthogonal to $S$ if and only if there exists a unit vector $ξ\in {\mathscr{H}}_0$ such that $\|T\|ξ= |T|ξ$ and $S^*Tξ= 0$. Finally, we introduce a new type of approximate orthogonality and investigate this notion in the setting of inner product $C^*$-modules.

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arXiv2016

Mappings preserving approximate orthogonality in Hilbert $C^*$-modules

We introduce a notion of approximate orthogonality preserving mappings between Hilbert $C^*$-modules. We define the concept of $(δ, \varepsilon)$-orthogonality preserving mapping and give some sufficient conditions for a linear mapping to be $(δ, \varepsilon)$-orthogonality preserving. In particular, if $\mathscr{E}$ is a full Hilbert $\mathscr{A}$-module with $\mathbb{K}(\mathscr{H})\subseteq \mathscr{A} \subseteq \mathbb{B}(\mathscr{H})$ and $T, S:\mathscr{E}\longrightarrow \mathscr{E}$ are two linear mappings satisfying $\big|\langle Sx, Sy\rangle\big| = \|S\|^2\,|\langle x, y\rangle|$ for all $x, y\in \mathscr{E}$ and $\|T - S\| \leq θ\|S\|$, then we show that $T$ is a $(δ, \varepsilon)$-orthogonality preserving mapping. We also prove whenever $\mathbb{K}(\mathscr{H})\subseteq \mathscr{A} \subseteq \mathbb{B}(\mathscr{H})$ and $T: \mathscr{E} \longrightarrow \mathscr{F}$ is a nonzero $\mathscr{A}$-linear $(δ, \varepsilon)$-orthogonality preserving mapping between $\mathscr{A}$-modules, then $$\big\|\langle Tx, Ty\rangle - \|T\|^2\langle x, y\rangle\big\|\leq \frac{4(\varepsilon - δ)}{(1 - δ)(1 + \varepsilon)} \|Tx\|\,\|Ty\|\qquad (x, y\in \mathscr{E}).$$ As a result, we present some characterizations of the orthogonality preserving mappings.

math.OA↗

arXiv2016

Conditionally positive definite kernels in Hilbert $C^*$-modules

We investigate the notion of conditionally positive definite in the context of Hilbert $C^*$-modules and present a characterization of the conditionally positive definiteness in terms of the usual positive definiteness. We give a Kolmogorov type representation of conditionally positive definite kernels in Hilbert $C^*$-modules. As a consequence, we show that a $C^*$-metric space $(S, d)$ is $C^*$-isometric to a subset of a Hilbert $C^*$-module if and only if $K(s,t)=-d(s,t)^2$ is a conditionally positive definite kernel. We also present a characterization of the order $K'\leq K$ between conditionally positive definite kernels.

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