arXiv subjects

Mohammad Sal Moslehian

Publications and source records attributed to Mohammad Sal Moslehian.

At least 55 records · Page 3Linked to original sources

arXiv2016

Unitarily invariant norm inequalities for elementary operators involving $G_{1}$ operators

In this paper, motivated by perturbation theory of operators, we present some upper bounds for $|||f(A)Xg(B)+ X|||$ in terms of $|||\,|AXB|+|X|\,|||$ and $|||f(A)Xg(B)- X|||$ in terms of $|||\,|AX|+|XB|\,|||$, where $A, B$ are $G_{1}$ operators, $|||\cdot|||$ is a unitarily invariant norm and $f, g$ are certain analytic functions. Further, we find some new upper bounds for the the Schatten $2$-norm of $f(A)X\pm Xg(B)$. Several special cases are discussed as well.

math.FA↗

arXiv2016

Grüss type inequalities for positive linear maps on $C^*$-algebras

Let $\mathcal{A}$ and $\mathcal{B}$ be two unital $C^*$-algebras and let for $C\in\mathcal{A},\ Γ_C=\{γ\in \mathbb{C} : \|C-γI\|=\inf_{α\in \mathbb{C}} \|C-αI\|\}$. We prove that if $Φ:\mathcal{A} \longrightarrow \mathcal{B}$ is a unital positive linear map, then \begin{eqnarray*} \big|Φ(AB)-Φ(A)Φ(B)\big| \leq \big\|Φ(|A^*-ζI|^2)\big\|^\frac{1}{2} \big[Φ(|B-ξI|^2)\big]^\frac{1}{2} \end{eqnarray*} for all $A,B\in\mathcal{A}, ζ\in Γ_A$ and $ξ\inΓ_B.$\\ In addition, we show that if $(\mathcal{A},τ)$ is a noncommutative probability space and $T \in \mathcal{A}$ is a density operator, then \begin{eqnarray*} \ \ \big|τ(TAB)-τ(TA)τ(TB)\big|\leq \|A-ζI\|_p\|B-ξI\|_q\|T\|_r \ \ (p,q\geq 4, r\geq 2) \end{eqnarray*} and \begin{eqnarray*} \big|τ(TAB)-τ(TA)τ(TB)\big|\leq \|A-ζI\|_p\|B-ξI\|_q\|T\| \ \ \ \ (p,q\geq 2)\ \ \ \ \ \end{eqnarray*} for every $A,B \in \mathcal{A}$ and $ζ\in Γ_A,ξ\in Γ_B$. Our results generalize the corresponding results for matrices to operators on spaces of arbitrary dimension.

math.OA↗

arXiv2016

Advanced refinements of Young and Heinz inequalities

In this article, we prove several multi-term refinements of Young type inequalities for both real numbers and operators improving several known results. Among other results, we prove \begin{eqnarray*} A\#_νB&+&\sum_{j=1}^{N}s_{j}(ν)\left(A\#_{α_j(ν)}B+A\#_{2^{1-j}+α_j(ν)}B-2A\#_{2^{-j}+α_j(ν)}B\right)\leq A\nabla_νB, \end{eqnarray*} for the positive operators $A$ and $B$, where $0\leq ν\leq 1, N\in\mathbb{N}$ and $α_j(ν)$ is a certain function. Moreover, some new Heinz type inequalities involving the Hilbert-Schmidt norm are established.

math.FA↗

arXiv2015

KSGNS type construction for $α$-completely positive maps on Krein $C^{\ast}$-modules

In this paper, we investigate $Φ$-maps associated to a certain type of $α$-completely positive maps. We then prove a KSGNS (Kasparov--Stinespring--Gel'fand--Naimark--Segal) type theorem for $α$-completely positive maps on Krein $C^*$-modules and show that the minimal KSGNS construction is unique up to unitary equivalence. We also establish a covariant version of the KSGNS type theorem for a covariant $α$-completely positive map and study the structure of minimal covariant KSGNS constructions.

math.OA↗

arXiv2015

Norm-parallelism in the geometry of Hilbert $C^*$-modules

Utilizing the Birkhoff--James orthogonality, we present some characterizations of the norm-parallelism for elements of $\mathbb{B}(\mathscr{H})$ defined on a finite dimensional Hilbert space, elements of a Hilbert $C^*$-module over the $C^*$-algebra of compact operators and elements of an arbitrary $C^*$-algebra. We also consider the characterization of norm parallelism problem for operators on a finite dimensional Hilbert space when the operator norm is replaced by the Schatten $p$-norm. Some applications and generalizations are discussed for certain elements of a Hilbert $C^*$-module.

math.OA↗

arXiv2015

Cartesian decomposition and Numerical radius inequalities

We show that if $T=H+iK$ is the Cartesian decomposition of $T\in \mathbb{B(\mathscr{H})}$, then for $α,β\in \mathbb{R}$, $\sup_{α^{2}+β^{2}=1}\Vert αH+βK\Vert =w(T)$. We then apply it to prove that if $A,B,X\in \mathbb{B(\mathscr{H})}$ and $0\leq mI\leq X$, then \begin{align*} m\Vert \mbox{Re}(A)-\mbox{Re}(B)\Vert & \leq w(\mbox{Re}(A)X-X\mbox{Re}(B)) \\ & \leq \frac{1}{2}\sup_{θ\in \mathbb{R}}\left\Vert (AX-XB)+e^{iθ}(XA-BX)\right\Vert \\ & \leq \frac{\Vert AX-XB\Vert +\Vert XA-BX\Vert }{2}, \end{align*} where $\mbox{Re}(T)$ denotes the real part of an operator $T$. A refinement of the triangle inequality is also shown.

math.FA↗

arXiv2015

Inequalities for operator space numerical radius of $2\times 2$ block matrices

In this paper, we study the relationship between operator space norm and operator space numerical radius on the matrix space $\mathcal{M}_n(X)$, when $X$ is a numerical radius operator space. Moreover, we establish several inequalities for operator space numerical radius and the maximal numerical radius norm of $2\times 2$ operator matrices and their off-diagonal parts. One of our main results states that if $(X, (O_n))$ is an operator space, then \begin{align*} \frac12\max\big(W_{\max}(x_1+x_2)&, W_{\max}(x_1-x_2) \big)\\ &\le W_{\max}\Big(\begin{bmatrix} 0 & x_1 \\ x_2 & 0 \end{bmatrix}\Big)\\ &\hspace{1.5cm}\le \frac12\left(W_{\max}(x_1+x_2)+ W_{\max}(x_1-x_2) \right) \end{align*} for all $x_1, x_2\in \mathcal{M}_n(X)$.

math.OA↗

arXiv2015

Angle Preserving Mappings

In this paper, we give some characterizations of orthogonality preserving mappings between inner product spaces. Furthermore, we study the linear mappings that preserve some angles. One of our main results states that if $\mathcal{X}, \mathcal{Y}$ are real inner product spaces and $θ\in(0, π)$, then an injective nonzero linear mapping $T:\mathcal{X}\longrightarrow \mathcal{Y}$ is a similarity whenever (i) $x\undersetθ{\angle} y\, \Leftrightarrow \,Tx\undersetθ{\angle} Ty$ for all $x, y\in \mathcal{X}$; (ii) for all $x, y\in \mathcal{X}$, $\|x\|=\|y\|$ and $x\undersetθ{\angle} y$ ensure that $\|Tx\|=\|Ty\|$. We also investigate orthogonality preserving mappings in the setting of inner product $C^{*}$-modules. Another result shows that if $\mathbb{K}(\mathscr{H})\subseteq\mathscr{A}\subseteq\mathbb{B}(\mathscr{H})$ is a $C^{*}$-algebra and $T\,:\mathscr{E}\longrightarrow \mathscr{F}$ is an $\mathscr{A}$-linear mapping between inner product $\mathscr{A}$-modules, then $T$ is orthogonality preserving if and only if $|x|\leq|y|\, \Rightarrow \,|Tx|\leq|Ty|$ for all $x, y\in \mathscr{E}$.

math.FA↗

arXiv2015

What to do to have your paper rejected!

We aim to highlight certain points and considerations for graduate students and young researchers, which should be avoided in submissions to good research journals. Observing these remarks could substantially decrease the probability of rejections of papers.

math.HO↗

arXiv2015

Squaring operator Pólya--Szegö and Diaz--Metcalf type inequalities

We square operator Pólya--Szegö and Diaz--Metcalf type inequalities as follows: If operator inequalities $0<m_{1}^{2} \leq A\leq M_{1}^{2}$ and $0<m_{2}^{2}\leq B\leq M_{2}^{2}$ hold for some positive real numbers $m_{1}\leq M_{1}$ and $m_{2}\leq M_{2}$, then for every unital positive linear map $Φ$ the following inequalities hold: \begin{eqnarray*} (Φ(A)\sharpΦ(B))^2 &\leq&\left(\frac{M_1M_2 + m_1m_2}{2\sqrt{M_1M_2m_1m_2}}\right)^4Φ(A\sharp B)^{2} \end{eqnarray*} and \begin{eqnarray*} \left( \frac{M_2m_2}{M_1m_1}Φ(A) + Φ(B) \right)^2 \leq \left( \frac{(M_1m_1(M_2^2 + m_2^2) + M_2m_2(M_1^2 + m_1^2))^2}{8\sqrt{M_2M_1m_1m_2} M_1^2m_1^2M_2m_2} \right)^2Φ(A\sharp B)^2\,. \end{eqnarray*}

math.FA↗

arXiv2014

Gruss inequality for some types of positive linear maps

Assuming a unitarily invariant norm $|||\cdot|||$ is given on a two-sided ideal of bounded linear operators acting on a separable Hilbert space, it induces some unitarily invariant norms $|||\cdot|||$ on matrix algebras $\mathcal{M}_n$ for all finite values of $n$ via $|||A|||=|||A\oplus 0|||$. We show that if $\mathscr{A}$ is a $C^*$-algebra of finite dimension $k$ and $Φ: \mathscr{A} \to \mathcal{M}_n$ is a unital completely positive map, then \begin{equation*} |||Φ(AB)-Φ(A)Φ(B)||| \leq \frac{1}{4} |||I_{n}|||\,|||I_{kn}||| d_A d_B \end{equation*} for any $A,B \in \mathscr{A}$, where $d_X$ denotes the diameter of the unitary orbit $\{UXU^*: U \mbox{ is unitary}\}$ of $X$ and $I_{m}$ stands for the identity of $\mathcal{M}_{m}$. Further we get an analogous inequality for certain $n$-positive maps in the setting of full matrix algebras by using some matrix tricks. We also give a Grüss operator inequality in the setting of $C^*$-algebras of arbitrary dimension and apply it to some inequalities involving continuous fields of operators.

math.FA↗

arXiv2014

Reverses of the Young inequality for matrices and operators

We present some reverse Young-type inequalities for the Hilbert-Schmidt norm as well as any unitarily invariant norm. Furthermore, we give some inequalities dealing with operator means. More precisely, we show that if $A, B\in {\mathfrak B}(\mathcal{H})$ are positive operators and $r\geq 0$, $A\nabla_{-r}B+2r(A\nabla B-A\sharp B)\leq A\sharp_{-r}B$ and prove that equality holds if and only if $A=B$. We also establish several reverse Young-type inequalities involving trace, determinant and singular values. In particular, we show that if $A, B$ are positive definite matrices and $r\geq 0$, then $\label{reverse_trace} \mathrm{tr}((1+r)A-rB)\leq \mathrm{tr}|A^{1+r}B^{-r} |-r(\sqrt{\mathrm{tr} A} - \sqrt{\mathrm{tr} B})^{2}$.

math.FA↗

arXiv2014

Some generalized numerical radius inequalities for Hilbert space operators

We generalize several inequalities involving powers of the numerical radius for product of two operators acting on a Hilbert space. For any $A, B, X\in \mathbb{B}(\mathscr{H})$ such that $A,B$ are positive, we establish some numerical radius inequalities for $A^αXB^α$ and $A^αX B^{1-α}\,\,(0 \leq α\leq 1)$ and Heinz means under mild conditions.

math.FA↗

arXiv2014

Noncommutative martingale concentration inequalities

We establish an Azuma type inequality under a Lipshitz condition for martingales in the framework of noncommutative probability spaces and apply it to deduce a noncommutative Heoffding inequality as well as a noncommutative McDiarmid type inequality. We also provide a noncommutative Azuma inequality for noncommutative supermartingales in which instead of a fixed upper bound for the variance we assume that the variance is bounded above by a linear function of variables. We then employ it to deduce a noncommutative Bernstein inequality and an inequality involving $L_p$-norm of the sum of a martingale difference.

math.OA↗

arXiv2014

Inequalities for sums of random variables in noncommutative probability spaces

In this paper, we establish an extension of a noncommutative Bennett inequality with a parameter $1\leq r\leq2$ and use it together with some noncommutative techniques to establish a Rosenthal inequality. We also present a noncommutative Hoeffding inequality as follows: Let $(\mathfrak{M}, τ)$ be a noncommutative probability space, $\mathfrak{N}$ be a von Neumann subalgebra of $\mathfrak{M}$ with the corresponding conditional expectation $\mathcal{E}_{\mathfrak{N}}$ and let subalgebras $\mathfrak{N}\subseteq\mathfrak{A}_j\subseteq\mathfrak{M}\,\,(j=1, \cdots, n)$ be successively independent over $\mathfrak{N}$. Let $x_j\in\mathfrak{A}_j$ be self-adjoint such that $a_j\leq x_j\leq b_j$ for some real numbers $a_j o$ it holds that \begin{eqnarray*} {\rm Prob}\left(\left|\sum_{j=1}^n x_j-nμ\right|\geq t\right)\leq 2 \exp\left\{\frac{-2t^2}{\sum_{j=1}^n(b_j-a_j)^2}\right\}. \end{eqnarray*}

math.OA↗

arXiv2014

Chebyshev type inequalities for Hilbert space operators

We establish several operator extensions of the Chebyshev inequality. The main version deals with the Hadamard product of Hilbert space operators. More precisely, we prove that if $\mathscr{A}$ is a $C^*$-algebra, $T$ is a compact Hausdorff space equipped with a Radon measure $μ$, $α: T\rightarrow [0, +\infty)$ is a measurable function and $(A_t)_{t\in T}, (B_t)_{t\in T}$ are suitable continuous fields of operators in ${\mathscr A}$ having the synchronous Hadamard property, then \begin{align*} \int_{T} α(s) dμ(s)\int_{T}α(t)(A_t\circ B_t) dμ(t)\geq\left(\int_{T}α(t) A_t dμ(t)\right)\circ\left(\int_{T}α(s) B_s dμ(s)\right). \end{align*} We apply states on $C^*$-algebras to obtain some versions related to synchronous functions. We also present some Chebyshev type inequalities involving the singular values of positive $n\times n$ matrices. Several applications are given as well.

math.FA↗

arXiv2014

Exact and approximate operator parallelism

Extending the notion of parallelism we introduce the concept of approximate parallelism in normed spaces and then substantially restrict ourselves to the setting of Hilbert space operators endowed with the operator norm. We present several characterizations of the exact and approximate operator parallelism in the algebra $\mathbb{B}(\mathscr{H})$ of bounded linear operators acting on a Hilbert space $\mathscr{H}$. Among other things, we investigate the relationship between approximate parallelism and norm of inner derivations on $\mathbb{B}(\mathscr{H})$. We also characterize the parallel elements of a $C^*$-algebra by using states. Finally we utilize the linking algebra to give some equivalence assertions regarding parallel elements in a Hilbert $C^*$-module.

math.FA↗

arXiv2014

Reverses and variations of Heinz inequality

Let $A, B$ be positive definite $n\times n$ matrices. We present several reverse Heinz type inequalities, in particular \begin{align*} \|AX+XB\|_2^2+ 2(ν-1) \|AX-XB\|_2^2\leq \|A^νXB^{1-ν}+A^{1-ν}XB^ν\|_2^2, \end{align*} where $X$ is an arbitrary $n \times n$ matrix, $\|\cdot\|_2$ is Hilbert-Schmidt norm and $ν>1$. We also establish a Heinz type inequality involving the Hadamard product of the form \begin{align*} 2|||A^{1\over2}\circ B^{1\over2}|||\leq|||A^{s}\circ B^{1-t}+A^{1-s}\circ B^{t}||| \leq\max\{|||(A+B)\circ I|||,|||(A\circ B)+I|||\}, \end{align*} in which $s, t\in [0,1]$ and $|||\cdot|||$ is a unitarily invariant norm.

math.FA↗