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Mohsen Kian

Publications and source records attributed to Mohsen Kian.

31 records · Page 2Linked to original sources
arXiv2020

Klein's trace inequality and superquadratic trace functions

We show that if $f$ is a non-negative superquadratic function, then $A\mapsto\mathrm{Tr}f(A)$ is a superquadratic function on the matrix algebra. In particular, \begin{align*} \tr f\left( {\frac{A + B}{2}} \right) +\tr f\left(\left| {\frac{A - B}{2}}\right|\right) \leq \frac{{\tr {f\left( A \right)} + \tr {f\left( B \right)} }}{2} \end{align*} holds for all positive matrices $A,B$. In addition, we present a Klein's inequality for superquadratic functions as $$ \mathrm{Tr}[f(A)-f(B)-(A-B)f'(B)]\geq \mathrm{Tr}[f(|A-B|)] $$ for all positive matrices $A,B$. It gives in particular an improvement of the Klein's inequality for non-negative convex function. As a consequence, some variants of the Jensen trace inequality for superquadratic functions have been presented.

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arXiv2020

Asymmetric Choi--Davis inequalities

Let $Φ$ be a unital positive linear map and let $A$ be a positive invertible operator. We prove that there exist partial isometries $U$ and $V$ such that \[ |Φ(f(A))Φ(A)Φ(g(A))|\leq U^*Φ(f(A)Ag(A))U \] and \[\left|Φ\left(f(A)\right)^{-r}Φ(A)^rΦ\left(g(A)\right)^{-r}\right|\leq V^*Φ\left(f(A)^{-r}A^rg(A)^{-r}\right)V\] hold under some mild operator convex conditions and some positive numbers $r$. Further, we show that if $f^2$ is operator concave, then $$ |Φ(f(A))Φ(A)|\leq Φ(Af(A)).$$ In addition, we give some counterparts to the asymmetric Choi--Davis inequality and asymmetric Kadison inequality. Our results extend some inequalities due to Bourin--Ricard and Furuta.

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arXiv2019

Variants of Ando--Hiai type inequalities for deformed means and applications

For an $n$-tuple of positive invertible operators on a Hilbert space, we present some variants of Ando--Hiai type inequalities for deformed means from an $n$-variable operator mean by an operator mean, which is related to the information monotonicity of a certain unital positive linear map. As an application, we investigate the monotonicity of the power mean from the deformed mean in terms of the generalized Kantorovich constants under the operator order. Moreover, we improve the norm inequality for the operator power means related to the Log-Euclidean mean in terms of the Specht ratio.

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arXiv2019

Matricial Radius: A Relation of Numerical radius with Matricial Range

It has been shown that if $T$ is a complex matrix, then {\small\begin{align*} ω(T)&=\frac{1}{n}\sup\left\{|\mathrm{Tr}\ X|;\ X\in W^n(T)\right\}\\ &=\frac{1}{n}\sup\left\{\|X\|_1;\ X\in W^n(T)\right\}\\ &= \sup\left\{ ω(X);\ X\in W^n(T)\right\} \end{align*} } where $n$ is a positive integer, $ω(T)$ is the numerical radius and $W^n(T)$ is the $n$'th matricial range of $T$.

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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.

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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.

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arXiv2018

Interpolating operator Jensen-type inequalities for log-convex and superquadratic functions

Motivated by some recently established operator Jensen-type inequalities related to a usual convexity, in the present paper we derive several more accurate operator Jensen-type inequalities for certain subclasses of convex functions. More precisely, we obtain interpolating series of Jensen-type inequalities for log-convex and non-negative superquadratic functions. In particular, we obtain the corresponding refinements of the Jensen-Mercer operator inequality for such classes of functions.

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arXiv2016

C*-Convexity of Norm Unit Balls

We determine those norms on B(H) whose unit ball is C*-convex. We call them M-norms and show that the class of M-norms less than a given norm enjoys a maximum element. These minimum and maximum elements will be determined in some cases. Finally, we give a constructive result to obtain M-norms on B(H).

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arXiv2015

Epigraph of Operator Functions

It is known that a real function $f$ is convex if and only if the set $$\mathrm{E}(f)=\{(x,y)\in\mathbb{R}\times\mathbb{R};\ f(x)\leq y\},$$ the epigraph of $f$ is a convex set in $\mathbb{R}^2$. We state an extension of this result for operator convex functions and $C^*$-convex sets as well as operator log-convex functions and $C^*$-log-convex sets. Moreover, the $C^*$-convex hull of a Hermitian matrix has been represented in terms of its eigenvalues.

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arXiv2014

Operator log-convex functions and f-divergence functional

We present a characterization of operator log-convex functions by using positive linear mappings. Moreover, we study the non-commutative f-divergence functional of operator log-convex functions. In particular, we prove that f is operator log-convex if and only if the non-commutative f-divergence functional is operator log-convex in its first variable and operator convex in its second variable.

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arXiv2013

Non-commutative f-divergence functional

We introduce the non-commutative $f$-divergence functional $Θ(\widetilde{A},\widetilde{B}):=\int_TB_t^{\frac{1}{2}}f\left(B_t^{-\frac{1}{2}} A_tB_t^{-\frac{1}{2}}\right)B_t^{\frac{1}{2}}dμ(t)$ for an operator convex function $f$, where $\widetilde{A}=(A_t)_{t\in T}$ and $\widetilde{B}=(B_t)_{t\in T}$ are continuous fields of Hilbert space operators and study its properties. We establish some relations between the perspective of an operator convex function $f$ and the non-commutative $f$-divergence functional. In particular, an operator extension of Csiszár's result regarding $f$-divergence functional is presented. As some applications, we establish a refinement of the Choi--Davis--Jensen operator inequality, obtain some unitarily invariant norm inequalities and give some results related to the Kullback--Leibler distance.

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