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M. S. Moslehian

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

At least 55 records · Page 3Linked to original sources
arXiv2013

Operator Entropy Inequalities

In this paper we investigate a notion of relative operator entropy, which develops the theory started by J.I. Fujii and E. Kamei [Math. Japonica 34 (1989), 341--348]. For two finite sequences $\mathbf{A}=(A_1,...,A_n)$ and $\mathbf{B}=(B_1,...,B_n)$ of positive operators acting on a Hilbert space, a real number $q$ and an operator monotone function $f$ we extend the concept of entropy by $$ S_q^f(\mathbf{A}|\mathbf{B}):=\sum_{j=1}^nA_j^{1/2}(A_j^{-1/2}B_jA_j^{-1/2})^qf(A_j^{-1/2}B_jA_j^{-1/2})A_j^{1/2}\,, $$ and then give upper and lower bounds for $S_q^f(\mathbf{A}|\mathbf{B})$ as an extension of an inequality due to T. Furuta [Linear Algebra Appl. 381 (2004), 219--235] under certain conditions. Afterwards, some inequalities concerning the classical Shannon entropy are drawn from it.

math.FA↗
arXiv2013

Further refinements of the Heinz inequality

The celebrated Heinz inequality asserts that $ 2|||A^{1/2}XB^{1/2}|||\leq |||A^νXB^{1-ν}+A^{1-ν}XB^ν|||\leq |||AX+XB|||$ for $X \in \mathbb{B}(\mathscr{H})$, $A,B\in \+$, every unitarily invariant norm $|||\cdot|||$ and $ν\in [0,1]$. In this paper, we present several improvement of the Heinz inequality by using the convexity of the function $F(ν)=|||A^νXB^{1-ν}+A^{1-ν}XB^ν|||$, some integration techniques and various refinements of the Hermite--Hadamard inequality. In the setting of matrices we prove that \begin{eqnarray*} &&\hspace{-0.5cm}\left|\left|\left|A^{\frac{α+β}{2}}XB^{1-\frac{α+β}{2}}+A^{1-\frac{α+β}{2}}XB^{\frac{α+β}{2}}\right|\right|\right|\leq\frac{1}{|β-α|} \left|\left|\left|\int_α^β\left(A^νXB^{1-ν}+A^{1-ν}XB^ν\right)dν\right|\right|\right|\nonumber\\ &&\qquad\qquad\leq \frac{1}{2}\left|\left|\left|A^αXB^{1-α}+A^{1-α}XB^α+A^βXB^{1-β}+A^{1-β}XB^β\right|\right|\right|\,, \end{eqnarray*} for real numbers $α, β$.

math.FA↗
arXiv2012

Moore-Penrose inverse of Gram operator on Hilbert C*-modules

Let $t$ be a regular operator between Hilbert $C^*$-modules and $t^†$ be its Moore-Penrose inverse. We investigate the Moore-Penrose invertibility of the Gram operator $t^*t$. More precisely, we study some conditions ensuring that $t^† = (t^* t)^† t^*= t^* (t t^*)^†$ and $(t^*t)^†=t^†t^{* †}$ hold. As an application, we get some results for densely defined closed operators on Hilbert $C^*$-modules over $C^*$-algebras of compact operators.

math.FA↗
arXiv2012

A General Double Inequality Related to Operator Means and Positive Linear Maps

Let $A,B\in \mathbb{B}(\mathscr{H})$ be such that $0<b_{1}I \leq A \leq a_{1}I$ and $0<b_{2}I \leq B \leq a_{2}I$ for some scalars $0<b_{i}< a_{i},\;\; i=1,2$ and $Φ:\mathbb{B}(\mathscr{H})\rightarrow\mathbb{B}(\mathscr{K})$ be a positive linear map. We show that for any operator mean $σ$ with the representing function $f$, the double inequality $$ ω^{1-α}(Φ(A)#_αΦ(B))\le (ωΦ(A))\nabla_αΦ(B)\leq \fracαμΦ(AσB) $$ holds, where $μ=\frac{a_{1}b_{1}(f(b_{2}a_{1}^{-1})-f(a_{2}b_{1}^{-1}))}{b_{1}b_{2}-a_{1}a_{2}}, $ $ν=\frac{a_{1}a_{2}f(b_{2}a_{1}^{-1})-b_{1}b_{2}f(a_{2}b_{1}^{-1})}{a_{1}a_{2}-b_{1}b_{2}}, $ $ω=\frac{αν}{(1-α)μ}$ and $#_α$ ($\nabla_α$, resp.) is the weighted geometric (arithmetic, resp.) mean for $α\in (0,1)$. As applications, we present several generalized operator inequalities including Diaz--Metcalf and reverse Ando type inequalities. We also give some related inequalities involving Hadamard product and operator means.

math.FA↗
arXiv2012

On the binary relation $\leq_u$ on self-adjoint Hilbert space operators

Given self-adjoint operators $A, B\in\mathbb{B}(\mathscr{H})$ it is said $A\leq_uB$ whenever $A\leq U^*BU$ for some unitary operator $U$. We show that $A\leq_u B$ if and only if $f(g(A)^r)\leq_uf(g(B)^r)$ for any increasing operator convex function $f$, any operator monotone function $g$ and any positive number $r$. We present some sufficient conditions under which if $B\leq A\leq U^*BU$, then $B=A=U^*BU$. Finally we prove that if $A^n\leq U^\ast A^nU$ for all $n\in\mathbb{N}$, then $A=U^\ast AU$.

math.OA↗
arXiv2012

Quasi-representations of Finsler modules over C*-algebras

We show that every Finsler module over a $C^*$-algebra has a quasi-representation into the Banach space $\mathbb{B}(\mathscr{H},\mathscr{K})$ of all bounded linear operators between some Hilbert spaces $\mathscr{H}$ and $\mathscr{K}$. We define the notion of completely positive $φ$-morphism and establish a Stinespring type theorem in the framework of Finsler modules over $C^*$-algebras. We also investigate the nondegeneracy and the irreducibility of quasi-representations.

math.OA↗
arXiv2011

Non-commutative Callebaut inequality

We present an operator version of the Callebaut inequality involving the interpolation paths and apply it to the weighted operator geometric means. We also establish a matrix version of the Callebaut inequality and as a consequence obtain an inequality including the Hadamard product of matrices.

math.FA↗
arXiv2011

Fuglede-Putnam type theorems via the Aluthge transform

Let $A=U|A|$ and $B=V|B|$ be the polar decompositions of $A\in \mathbb{B}(\mathscr{H}_1)$ and $B\in \mathbb{B}(\mathscr{H}_2)$ and let $Com(A,B)$ stand for the set of operators $X\in\mathbb{B}(\mathscr{H}_2,\mathscr{H}_1)$ such that $AX=XB$. A pair $(A,B)$ is said to have the FP-property if $Com(A,B)\subseteqCom(A^\ast,B^\ast)$. Let $\tilde{C}$ denote the Aluthge transform of a bounded linear operator $C$. We show that (i) if $A$ and $B$ are invertible and $(A,B)$ has the FP-property, then so is $(\tilde{A},\tilde{B})$; (ii) if $A$ and $B$ are invertible, the spectrums of both $U$ and $V$ are contained in some open semicircle and $(\tilde{A},\tilde{B})$ has the FP-property, then so is $(A,B)$; (iii) if $(A,B)$ has the FP-property, then $Com(A,B)\subseteqCom(\tilde{A},\tilde{B})$, moreover, if $A$ is invertible, then $Com(A,B)=Com(\tilde{A},\tilde{B})$. Finally, if $Re(U|A|^{1\over2})\geq a>0$ and $Re(V|B|^{1\over2})\geq a>0$ and $X$ is an operator such that $U^* X=XV$, then we prove that $\|\tilde{A}^* X-X\tilde{B}\|_p\geq 2a\|\,|B|^{1\over2}X-X|B|^{1\over2}\|_p$ for any $1 \leq p \leq \infty$.

math.FA↗
arXiv2011

Around Operator Monotone Functions

We show that the symmetrized product $AB+BA$ of two positive operators $A$ and $B$ is positive if and only if $f(A+B)\leq f(A)+f(B)$ for all non-negative operator monotone functions $f$ on $[0,\infty)$ and deduce an operator inequality. We also give a necessary and sufficient condition for that the composition $f\circ g$ of an operator convex function $f$ on $[0,\infty)$ and a non-negative operator monotone function $g$ on an interval $(a,b)$ is operator monotone and give some applications.

math.FA↗
arXiv2011

Characterization of a generalized triangle inequality in normed spaces

For a normed linear space $(X,|\cdot|)$ and $p>0$ we characterize all $n$-tuples $(μ_1,...,μ_n)\in\mathbb{R}^{n}$ for which the generalized triangle inequality of the second type $$\|x_1+...+x_n\|^p\leq\frac{|x_1|^p}{μ_1}+...+\frac{|x_n|^p}{μ_n}$$ holds for any $x_1,...,x_n\in X$. We also characterize $(μ_1,...,μ_n)\in\mathbb{R}^{n}$ for which the reverse of the inequality above holds.

math.FA↗
arXiv2011

Some conditions implying normality of operators

Let $T\in\mathbb{B}(\mathscr{H})$ and $T=U|T|$ be its polar decomposition. We proved that (i) if $T$ is log-hyponormal or $p$-hyponormal and $U^n=U^\ast$ for some $n$, then $T$ is normal; (ii) if the spectrum of $U$ is contained in some open semicircle, then $T$ is normal if and only if so is its Aluthge transform $\widetilde{T}=|T|^{1\over2}U|T|^{1\over2}$.

math.FA↗
arXiv2011

A Diaz--Metcalf type inequality for positive linear maps and its applications

We present a Diaz--Metcalf type operator inequality as a reverse Cauchy-Schwarz inequality and then apply it to get the operator versions of Pólya-Szegö's, Greub-Rheinboldt's, Kantorovich's, Shisha-Mond's, Schweitzer's, Cassels' and Klamkin-McLenaghan's inequalities via a unified approach. We also give some operator Grüss type inequalities and an operator Ozeki-Izumino-Mori-Seo type inequality. Several applications are concluded as well.

math.FA↗
arXiv2011

A treatment of the Cauchy--Schwarz inequality in $C^*$-modules

We study the Cauchy--Schwarz and some related inequalities in a semi-inner product module over a $C^*$-algebra $\A$. The key idea is to consider a semi-inner product $\A$-module as a semi-inner product $\A$-module with respect to another semi-inner product. In this way, we improve some inequalities such as the Ostrowski inequality and an inequality related to the Gram matrix. The induced semi-inner products are also related to the the notion of covariance and variance. Furthermore, we obtain a sequence of nested inequalities that emerges from the Cauchy--Schwarz inequality. As a consequence, we derive some interesting operator-theoretical corollaries. In particular, we show that the sequence arising from our construction, when applied to a positive element of a $C^*$-algebra, converges to its pseudo-inverse.

math.OA↗
arXiv2011

Unitarily invariant norm inequalities for operators

We present several operator and norm inequalities for Hilbert space operators. In particular, we prove that if $A_{1},A_{2},...,A_{n}\in {\mathbb B}({\mathscr H})$, then \[|||A_{1}A_{2}^{*}+A_{2}A_{3}^{*}+...+A_{n}A_{1}^{*}|||\leq|||\sum_{i=1}^{n}A_{i}A_{i}^{*}|||,\] for all unitarily invariant norms. We also show that if $A_{1},A_{2},A_{3},A_{4}$ are projections in ${\mathbb B}({\mathscr H})$, then &&|||(\sum_{i=1}^{4}(-1)^{i+1}A_{i})\oplus0\oplus0\oplus0|||&\leq&|||(A_{1}+|A_{3}A_{1}|)\oplus (A_{2}+|A_{4}A_{2}|)\oplus(A_{3}+|A_{1}A_{3}|)\oplus(A_{4}+|A_{2}A_{4}|)||| for any unitarily invariant norm.

math.FA↗
arXiv2010

Dunkl--Williams inequality for operators \\ associated with $p$-angular distance

We present several operator versions of the Dunkl--Williams inequality with respect to the $p$-angular distance for operators. More precisely, we show that if $A, B \in \mathbb{B}(\mathscr{H})$ such that $|A|$ and $|B|$ are invertible, $\frac{1}{r}+\frac{1}{s}=1\,\,(r>1)$ and $p\in\mathbb{R}$, then \begin{equation*} |A|A|^{p-1}-B|B|^{p-1}|^{2} \leq |A|^{p-1}(r|A-B|^{2}+s||A|^{1-p}|B|^{p}-|B||^2)|A|^{p-1}.%\nonumber \end{equation*} In the case that $0 0$, then $$|(U|A|^{p}-V|B|^{p})|A|^{1-p}|^{2}\leq (1+t)|A-B|^{2}+(1+\frac{1}{t})||B|^{p}|A|^{1-p}-|B||^2 \,.$$ We obtain several equivalent conditions, when the case of equalities hold.

math.OA↗
arXiv2009

Reverse triangle inequality in Hilbert $C^*$-modules

We prove several versions of reverse triangle inequality in Hilbert $C^*$-modules. We show that if $e_1, ..., e_m$ are vectors in a Hilbert module ${\mathfrak X}$ over a $C^*$-algebra ${\mathfrak A}$ with unit 1 such that $ =0 (1\leq i\neq j \leq m)$ and $\|e_i\|=1 (1\leq i\leq m)$, and also $r_k,ρ_k\in\Bbb{R} (1\leq k\leq m)$ and $x_1, ..., x_n\in {\mathfrak X}$ satisfy $$0\leq r_k^2 \|x_j\|\leq {Re}< r_ke_k,x_j> ,\quad0\leq ρ_k^2 \|x_j\| \leq {Im}< ρ_ke_k,x_j> ,$$ then [\sum_{k=1}^m(r_k^2+ρ_k^2)]^{1/2}\sum_{j=1}^n \|x_j\|\leq\|\sum_{j=1}^nx_j\|, and the equality holds if and only if \sum_{j=1}^n x_j=\sum_{j=1}^n\|x_j\|\sum_{k=1}^m(r_k+iρ_k)e_k .

math.FA↗
arXiv2009

$(σ,τ)$-amenability of $C^*$-algebras

Suppose that ${\mathcal A}$ is an algebra, $σ,τ:{\mathcal A}\to{\mathcal A}$ are two linear mappings such that both $σ({\mathcal A})$ and $τ({\mathcal A})$ are subalgebras of ${\mathcal A}$ and ${\mathcal X}$ is a $\big(τ({\mathcal A}),σ({\mathcal A})\big)$-bimodule. A linear mapping $D:{\mathcal A}\to {\mathcal X}$ is called a $(σ,τ)$-derivation if $D(ab)=D(a)\cdotσ(b)+τ(a)\cdot D(b) (a,b\in {\mathcal A})$. A $(σ,τ)$-derivation $D$ is called a $(σ,τ)$-inner derivation if there exists an $x\in{\mathcal X}$ such that $D$ is of the form either $D_x^-(a)=x\cdot σ(a)-τ(a)\cdot x (a\in {\mathcal A})$ or $D_x^ +(a)=x\cdot σ(a)+τ(a)\cdot x (a\in {\mathcal A})$. A Banach algebra ${\mathcal A}$ is called $(σ,τ)$-amenable if every $(σ,τ)$-derivation from ${\mathcal A}$ into a dual Banach $\big(τ({\mathcal A}),σ({\mathcal A})\big)$-bimodule is $(σ,τ)$-inner. Studying some general algebraic aspects of $(σ,τ)$-derivations, we investigate the relation between amenability and $(σ,τ)$-amenability of Banach algebras in the case when $σ, τ$ are homomorphisms. We prove that if $\mathfrak A$ is a $C^*$-algebra and $σ, τ$ are *-homomorphisms with $\ker(σ)=\ker(τ)$, then ${\mathfrak A}$ is $(σ, τ)$-amenable if and only if $σ({\mathfrak A})$ is amenable

math.OA↗