arXiv subjects

Mojtaba Bakherad

Publications and source records attributed to Mojtaba Bakherad.

At least 19 recordsLinked to original sources

arXiv2023

Further norm and numerical radius inequalities for sum of Hilbert space operators

Let ${\mathbb B}(\mathscr H)$ denote the set of all bounded linear operators on a complex Hilbert space ${\mathscr H}$. In this paper, we present some norm inequalities for sums of operators which are a generalization of some recent results. Among other inequalities, it is shown that if $S, T\in {\mathbb B}({\mathscr H})$ are normal operators, then \begin{eqnarray*} \left\Vert S+T\right\Vert \leq \frac{1}{2}(\left\Vert S\right\Vert+\left\Vert T\right\Vert)+\frac{1}{2}\min_{t>0}\sqrt{ (\left\Vert S \right\Vert-\left\Vert T\right\Vert)^2+ \left\Vert \frac{1}{t} f_1(\vert S \vert)g_1(\vert T\vert)+tf_2(\vert S \vert)g_2(\vert T\vert) \right\Vert^2}, \end{eqnarray*} where $f_1,f_2,g_1,g_2$ are non-negative continuous functions on $[0,\infty )$, in which $f_1(x)f_2(x)=x$ and $g_1(x)g_2(x)=x\,\,(x\geq 0)$. Moreover, it is shown several inequalities for the numerical radius.

math.FA↗

arXiv2023

Some extensions of Berezin number inequalities on operators

In this paper, we establish some upper bounds for Berezin number inequalities including of $2\times 2$ operator matrices and their off-diagonal parts. Among other inequalities, it is shown that if $T=\left[\begin{array}{cc} 0&X, Y&0 \end{array}\right]$, then \begin{align*} \textbf{ber}^{r}(T)\leq 2^{r-2}\left(\textbf{ber}(f^{2r}(|X|)+g^{2r}(|Y^*|))+\textbf{ber}(f^{2r}(|Y|)+g^{2r}(|X^*|))\right)\\ -2^{r-2} \inf_{\|(k_{λ_{1}},k_{λ_{2}})\|=1} η(k_{λ_{1}},k_{λ_{2}}), \end{align*} where $η(k_{λ_{1}}, k_{λ_{2}}) = \left(\left\langle(f^{2r}(|X|)+g^{2r}(|Y^*|)\right)k_{λ_{2}},k_{λ_{2}}\right\rangle^\frac{1}{2}-\left\langle \left(f^{2r}(|Y|)+g^{2r}(|X^*|)\right)k_{λ_{1}},k_{λ_{1}}\right\rangle^\frac{1}{2})^2$, $X, Y$ are bounded linear operators on a Hilbert space $\mathcal H=\mathcal H(Ω)$, $r\geq 1$ and $f$, $g$ are nonnegative continuous functions on $[0, \infty)$ satisfying the relation $f(t)g(t)=t\,(t\in[0, \infty))$.

math.FA↗

arXiv2018

Noncommutative Chebyshev inequality involving the Hadamard product

We present several operator extensions of the Chebyshev inequality for Hilbert space operators. The main version deals with the synchronous Hadamard property for Hilbert space operators. Among other inequalities, it is shown that if ${\mathfrak A}$ is a $C^*$-algebra, $T$ is a compact Hausdorff space equipped with a Radon measure $μ$ as a totaly order set, then \begin{align*} \int_{T} α(s) dμ(s)\int_{T}α(t)(A_t\circ B_t) dμ(t)\geq\Big{(}\int_{T}α(t) (A_tm_{r,α} B_t) dμ(t)\Big{)}\circ\Big{(}\int_{T}α(s) (A_sm_{r,1-α} B_s) dμ(s)\Big{)}, \end{align*} where $α\in[0,1]$, $r\in[-1,1]$ and $(A_t)_{t\in T}, (B_t)_{t\in T} $ are positive increasing fields in $\mathcal{C}(T,\mathfrak A)$.

math.FA↗

arXiv2018

Further refinements of generalized numerical radius inequalities for Hilbert space operators

In this paper, we show some refinements of generalized numerical radius inequalities involving the Young and Heinz inequalities. In particular, we present \begin{align*} w_{p}^{p}(A_{1}^{*}T_{1}B_{1},...,A_{n}^{*}T_{n}B_{n})\leq\frac{n^{1-\frac{1}{r}}}{2^{\frac{1}{r}}}\Big\|\sum_{i=1}^{n}[B_{i}^{*} f^{2}(|T_{i}|)B_{i}]^{rp}+[A_{i}^{*}g^{2}(|T_{i}^{*}|)A_{i}]^{rp}\Big\|^{\frac{1}{r}} -\inf_{\|x\|=1}η(x), \end{align*} where $T_{i}, A_{i}, B_{i} \in {\mathbb B}({\mathscr H})\,\,(1\leq i\leq n)$, $f$ and $g$ are nonnegative continuous functions on $[0, \infty)$ satisfying $f(t)g(t)=t$ for all $t\in [0, \infty)$, $p, r\geq 1$, $N\in {\mathbb N}$ and \begin{align*} η(x)= \frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{N} \Big(\sqrt[2^{j}]{ \langle (A_{i}^{*}g^{2}(|T_{i}^{*}|)A_{i})^{p}x, x\rangle^{2^{j-1}-k_{j}} \langle (B_{i}^{*} f^{2}(|T_{i}|)B_{i})^{p}x, x\rangle^{k_j}}\quad-\sqrt[2^{j}]{ \langle (B_{i}^{*}f^{2}(|T_{i}|)B_{i})^{p}x, x\rangle^{k_{j}+1} \langle (A_{i}^{*} g^{2}(|T_{i}^{*}|)A_{i})^{p}x, x\rangle^{2^{j-1}-k_{j}-1}}\Big)^{2}. \end{align*}

math.FA↗

arXiv2018

Improvements of Berezin number inequalities

In this paper, we generalize several Berezin number inequalities involving product of operators. For instance, we show that if $A, B$ are positive operators and $X$ is any operator, then \begin{align*} \textbf{ber}^{r}(H_α(A,B))&\leq\frac{\|X\|^{r}}{2}\textbf{ber}(A^{r}+B^{r})&\leq\frac{\|X\|^{r}}{2}\textbf{ber}(αA^{r}+(1-α)B^{r})+\textbf{ber}((1-α)A^{r}+αB^{r}), \end{align*} where $H_α(A,B)=\frac{A^αXB^{1-α}+A^{1-α} XB^α}{2}$, $0\leqα\leq1$ and $r\geq2$.

math.FA↗

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.

math.FA↗

arXiv2018

Berezin number inequalities for Hilbert space operators

In this paper, by using of the definition Berezin symbol, we show some Berezin number inequalities. Among other inequalities, it is shown that if $A, B, X\in{\mathbb{B}}(\mathscr H)$, then $$\mathbf{ber}(AX\pm XA)\leqslant \mathbf{ber}^{\frac{1}{2}}\left(A^*A+AA^*\right)\mathbf{ber}^{\frac{1}{2}}\left(X^*X+XX^*\right)$$ and $$\mathbf{ber}^2(A^*XB)\leqslant\|X\|^2\mathbf{ber}(A^*A)\mathbf{ber}(B^*B).$$

math.FA↗

arXiv2018

Some Berezin number inequalities for operator matrices

The Berezin symbol $\widetilde{A}$ of an operator $A$ acting on the reproducing kernel Hilbert space ${\mathscr H}={\mathscr H(}Ω)$ over some (non-empty) set is defined by $\widetilde{A}(λ)=\langle A\hat{k}_λ,\hat{k}_λ\rangle\,\,\,(λ\inΩ)$, where $\hat{k}_λ=\frac{{k}_λ}{\|{k}_λ\|}$ is the normalized reproducing kernel of ${\mathscr H}$. The Berezin number of operator $A$ is defined by $\mathbf{ber}(A) = \underset{λ\in Ω}{\sup} \big|\tilde{A}(λ)\big|=\underset{λ\in Ω}{\sup} \big|\langle A\hat{k}_λ, \hat{k}_λ\rangle\big|$. Moreover $\mathbf{ber}(A)\leqslant w(A)$ (numerical radius). In this paper, we present some Berezin number inequalities. Among other inequalities, it is shown that if $\mathbf{T}=\left[\begin{array}{cc} A&B C&D \end{array}\right]\in {\mathbb B}({\mathscr H(Ω_1)}\oplus{\mathscr H(Ω_2)})$, then \begin{align*} \mathbf{ber}(\mathbf{T}) \leqslant\frac{1}{2}\left( \mathbf{ber}(A)+ \mathbf{ber}(D)\right)+\frac{1}{2}\sqrt{\left( \mathbf{ber}(A)- \mathbf{ber}(D)\right)^2+(\|B\|+\|C\|)^2}. \end{align*}

math.FA↗

arXiv2018

Some generalized numerical radius inequalities involving Kwong functions

We prove several numerical radius inequalities involving positive semidefinite matrices via the Hadamard product and Kwong functions. Among other inequalities, it is shown that if $X$ is an arbitrary $n\times n$ matrix and $A,B$ are positive semidefinite, then \begin{align*} ω(H_{f,g}(A))\leq k\, ω(AX+XA), \end{align*} which is equivalent to \begin{align*} ω\big(H_{f,g}(A,B)\pm H_{f,g}(B,A)\big)\leq k'\,\left\{ω((A+B)X+X(A+B))+ω((A-B)X-X(A-B))\right\}, \end{align*} where $f$ and $g$ are two continuous functions on $(0,\infty)$ such that $h(t)={f(t)\over g(t)}$ is Kwong, $k=\max\left\{{f(λ)g(λ)\over λ}: {λ\inσ(A)}\right\}$ and $k'=\max\left\{{f(λ)g(λ)\over λ}: {λ\inσ(A)\cupσ(B)}\right\}$.

math.FA↗

arXiv2018

Unitarily invariant norm inequalities involving $G_1$ operators

In this paper, we present some upper bounds for unitarily invariant norms inequalities. Among other inequalities, we show some upper bounds for the Hilbert-Schmidt norm. In particular, we prove \begin{align*} \|f(A)Xg(B)\pm g(B)Xf(A)\|_2\leq \left\|\frac{(I+|A|)X(I+|B|)+(I+|B|)X(I+|A|)}{d_Ad_B}\right\|_2, \end{align*} where $A, B, X\in\mathbb{M}_n$ such that $A$, $B$ are Hermitian with $σ(A)\cupσ(B)\subset\mathbb{D}$ and $f, g$ are analytic on the complex unit disk $\mathbb{D}$, $g(0)=f(0)=1$, $\textrm{Re}(f)>0$ and $\textrm{Re}(g)>0$.

math.FA↗

arXiv2018

Improvements of some operator inequalities involving positive linear maps via the Kantorovich constant

We present some operator inequalities for positive linear maps that generalize and improve the derived results in some recent years. For instant, if $A$ and $B$ are positive operators and $m,m^{'},M,M^{'}$ are positive real numbers satisfying either one of the condition $ 0<m \leq B \leq m^{'} <M^{'} \leq A \leq M $ or $0<m \leq A \leq m^{'} <M^{'} \leq B \leq M$, then \begin{align*} Φ^{p} \big(A \nabla _{v} B+2 r Mm (A^{-1}\nabla B^{-1}- &A^{-1} \sharp B^{-1} )\big)\\ & \leq \left( \frac{K(h)}{ 4^{\frac{2}{p}-1} K^{r_{1}} \left( \sqrt {h^{'}}\right)} \right) ^{p} Φ^{p} (A \sharp_ν B) \end{align*} and \begin{align*} Φ^{p} \big(A \nabla _{v} B+2 r Mm (A^{-1}\nabla B^{-1}-& A^{-1} \sharp B^{-1} )\big) \\ &\leq \left( \frac{K(h)}{ 4^{\frac{2}{p}-1} K^{r_{1}}\left( \sqrt {h^{'}}\right)}\right) ^{p} (Φ(A) \sharp_ν Φ(B))^{p}, \end{align*} where $Φ$ is a positive unital linear map, $ 0 \leq ν\leq 1$, $p \geq 2,$ $r=\min\{ν,1-ν\},$ $h=\frac{M}{m},$ $h^{'}=\frac{M^{'}}{m^{'}}$, $K(h)=\frac{(1+h)^{2}}{4h}$ and $r_{1}=\min\{2r,1-2r\}.$ We also obtain a reverse of the Ando inequality for positive linear maps via the Kantorovich constant.

math.FA↗

arXiv2017

Some generalizations of the Aluthge transform of operators

Let $A = U |A|$ be the polar decomposition of $A$. The Aluthge transform of the operator $A$, denoted by $\tilde{A}$, is defined as $\tilde{A} =|A|^{\frac{1}{2}} U |A|^{\frac{1}{2}}$. In this paper, first we generalize the definition of Aluthge transform for non-negative continuous functions $f, g$ such that $f(x)g(x)=x\,\,(x\geq0)$. Then, by using of this definition, we get some numerical radius inequalities. Among other inequalities, it is shown that if $A$ is bounded linear operator on a complex Hilbert space ${\mathscr H}$, then \begin{equation*} h\left( w(A)\right) \leq \frac{1}{4}\left\Vert h\left( g^{2}\left( \left\vert A\right\vert \right) \right) +h\left( f^{2}\left( \left\vert A\right\vert \right) \right) \right\Vert +\frac{1}{2}h\left( w\left( \tilde{A}_{f,g}\right) \right) , \end{equation*} where $f, g$ are non-negative continuous functions such that $f(x)g(x)=x\,\,(x\geq 0)$, $h$ is a non-negative non-decreasing convex function on $[0,\infty )$ and $\tilde{A}_{f,g} =f(|A|) U g(|A|)$.

math.FA↗

arXiv2017

Numerical radius inequalities involving commutators of $G_{1}$ operators

We prove numerical radius inequalities involving commutators of $G_{1}$ operators and certain analytic functions. Among other inequalities, it is shown that if $A$ and $X$ are bounded linear operators on a complex Hilbert space, then \begin{equation*} w(f(A)X+X\bar{f}(A))\leq {\frac{2}{d_{A}^{2}}}w(X-AXA^{\ast }), \end{equation*} where $A$ is a $G_{1}$ operator with $σ(A)\subset \mathbb{D}$ and $f$ is analytic on the unit disk $\mathbb{D}$ such that $\textrm{Re}(f)>0$ and $f(0)=1$.

math.FA↗

arXiv2017

Extensions of interpolation between the arithmetic-geometric mean inequality for matrices

In this paper, we present some extensions of interpolation between the arithmetic-geometric means inequality. Among other inequalities, it is shown that if $A, B, X$ are $n\times n$ matrices, then \begin{align*} \|AXB^*\|^2\leq\|f_1(A^*A)Xg_1(B^*B)\|\,\|f_2(A^*A)Xg_2(B^*B)\|, \end{align*} where $f_1,f_2,g_1,g_2$ are non-negative continues functions such that $f_1(t)f_2(t)=t$ and $g_1(t)g_2(t)=t\,\,(t\geq0)$. We also obtain the inequality \begin{align*} \left|\left|\left|AB^*\right|\right|\right|^2\nonumber&\leq \left|\left|\left|p(A^*A)^{\frac{m}{p}}+ (1-p)(B^*B)^{\frac{s}{1-p}}\right|\right|\right|\,\left|\left|\left|(1-p)(A^*A)^{\frac{n}{1-p}}+ p(B^*B)^{\frac{t}{p}}\right|\right|\right|, \end{align*} in which $m,n,s,t$ are real numbers such that $m+n=s+t=1$, $|||\cdot|||$ is an arbitrary unitarily invariant norm and $p\in[0,1]$.

math.FA↗

arXiv2017

Some generalizations of numerical radius on off-diagonal part of $2\times 2$ operator matrices

We generalize several inequalities involving powers of the numerical radius for off-diagonal part of $2\times2$ operator matrices of the form $T=\left[\begin{array}{cc} 0&B, C&0 \end{array}\right]$, where $B, C$ are two operators. In particular, if $T=\left[\begin{array}{cc} 0&B, C&0 \end{array}\right]$, then we get \begin{align*} {1\over 2^{{3\over2}(r-1)}}\max\{ \| μ\|, \| η\| \} \leq w^{r}(T)\leq \frac{1}{2^{r+1}} \max\{ \| μ\|, \| η\| \}, \end{align*} where $r\geq 2$ and $ μ=|(C-B^{*})+i(C+B^{*})|^{r}+|(B^{*}-C)+i(C+B^{*})|^{r}$, $ η=|(B-C^{*})+i(B+C^{*})|^{r}+|(C^{*}-B)+i(B+C^{*})|^{r}$.

math.FA↗

arXiv2017

Upper bounds for numerical radius inequalities involving off-diagonal operator matrices

In this paper, we establish some upper bounds for numerical radius inequalities including of $2\times 2$ operator matrices and their off-diagonal parts. Among other inequalities, it is shown that if $T=\left[\begin{array}{cc} 0&X, Y&0 \end{array}\right]$, then \begin{align*} ω^{r}(T)\leq 2^{r-2}\left\|f^{2r}(|X|)+g^{2r}(|Y^*|)\right\|^\frac{1}{2}\left\|f^{2r}(|Y|)+g^{2r}(|X^*|)\right\|^\frac{1}{2} \end{align*} and \begin{align*} ω^{r}(T)\leq 2^{r-2}\left\|f^{2r}(|X|)+f^{2r}(|Y^*|)\right\|^\frac{1}{2}\left\|g^{2r}(|Y|)+g^{2r}(|X^*|)\right\|^\frac{1}{2}, \end{align*} where $X, Y$ are bounded linear operators on a Hilbert space ${\mathscr H}$, $r\geq 1$ and $f$, $g$ are nonnegative continuous functions on $[0, \infty)$ satisfying the relation $f(t)g(t)=t\,(t\in[0, \infty))$. Moreover, we present some inequalities involving the generalized Euclidean operator radius of operators $T_{1},\cdots,T_{n}$.

math.FA↗

arXiv2017

Some extensions of the Young and Heinz inequalities for Matrices

In this paper, we present some extensions of the Young and Heinz inequalities for the Hilbert-Schmidt norm as well as any unitarily invariant norm. Furthermore, we give some inequalities dealing with matrices. More precisely, for two positive semidefinite matrices $A$ and $B$ we show that \begin{align*} \Big\|A^νXB^{1-ν}+A^{1-ν}XB^ν\Big\|_{2}^{2}\leq\Big\|AX+XB\Big\|_{2}^{2}- 2r\Big\|AX-XB\Big\|_{2}^{2}-r_{0}\left(\Big\|A^{\frac{1}{2}}XB^{\frac{1}{2}}-AX\Big\|_{2}^{2}+ \Big\|A^{\frac{1}{2}}XB^{\frac{1}{2}}-XB\Big\|_{2}^{2}\right), \end{align*} where $X$ is an arbitrary $n\times n$ matrix, $0<ν\leq\frac{1}{2}$, $r=\min\{ν, 1-ν\}$ and $r_{0}=\min\{2r, 1-2r\}$.

math.FA↗

arXiv2017

Some extensions of the operator entropy type inequalities

In this paper, we establish some reverses of the operator entropy inequalities under certain conditions by using the Mond-Pečarić method. In particular, we present {\tiny \begin{align*} f&\left[\int_T(A_s\natural_{p+1}B_s)dμ(s)+t_0\left(I_{\mathscr H}-\int_TA_s\natural_pB_sdμ(s)\right)\right]-γ_ff(t_0)\left(I_{\mathscr H}-\int_TA_s\natural_pB_sdμ(s)\right)\nonumber\\ &\le γ_f\widetilde{S}_p^f(\mathbf{A}|\mathbf{B})\,, \end{align*}} where $T$ is a locally compact Hausdorff space and $μ$ is a Radon measure on $T$, $0<m A_s \leq B_s \leq M A_s\,\,(s\in T)$ for some positive real numbers $m, M$ such that $m<1<M$, $\int_TA_s=\int_TB_s=I_{\mathscr H}$, $f: (0,\infty) \to [0,\infty)$ be operator concave, $γ_f=\max\left\{\frac{f(t)}{μ_f t+ν_f}: m\leq t\leq M,μ_f=\frac{f(M)-f(m)}{M-m}, ν_f=\frac{Mf(m)-mf(M)}{M-m}\right\}$, $t_0\in[m,M]$, $p\in[0,1]$, and $$ \widetilde{S}_p^f(\mathbf{A}|\mathbf{B})=\int_TA_s^{\frac{1}{2}}\left(A_s^{-\frac{1}{2}}B_sA_s^{-\frac{1}{2}}\right)^p f\left(A_s^{-\frac{1}{2}}B_sA_s^{-\frac{1}{2}}\right)A_s^{\frac{1}{2}}dμ(s)\,. $$

math.FA↗