Showing papers by "Mojtaba Bakherad published in 2018"

Some Berezin Number Inequalities for Operator Matrices

Abstract: The Berezin symbol A of an operator A acting on the reproducing kernel Hilbert space H = H(Ω) over some (nonempty) set is defined by \(\tilde A(\lambda ) = \left\langle {A\hat k_\lambda ,\hat k_\lambda } \right\rangle \), λ ∈ Ω, where \(\hat k_\lambda = k_\lambda /\left\| {k_\lambda } \right\|\) is the normalized reproducing kernel of H. The Berezin number of the operator A is defined by \(ber(A) = \mathop {\sup }\limits_{\lambda \in \Omega } \left| {\tilde A(\lambda )} \right| = \mathop {\sup }\limits_{\lambda \in \Omega } \left| {\left\langle {A\hat k_\lambda ,\hat k_\lambda } \right\rangle } \right|\). Moreover, ber(A) ⩽ w(A) (numerical radius). We present some Berezin number inequalities. Among other inequalities, it is shown that if \(T = \left[ {\begin{array}{*{20}c} A & B \\ C & D \\ \end{array} } \right] \in \mathbb{B}(\mathcal{H}(\Omega _1 ) \oplus \mathcal{H}(\Omega _2 ))\), then $$ber(T) \leqslant \frac{1} {2}(ber(A) + ber(D)) + \frac{1} {2}\sqrt {(ber(A) - ber(D))^2 + \left( {\left\| B \right\| + \left\| C \right\|} \right)^2 } .$$

Some generalized numerical radius inequalities involving Kwong functions

Abstract: 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*} \omega(H_{f,g}(A))\leq k\, \omega(AX+XA), \end{align*} which is equivalent to \begin{align*} \omega\big(H_{f,g}(A,B)\pm H_{f,g}(B,A)\big)\leq k'\,\left\{\omega((A+B)X+X(A+B))+\omega((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, $k=\max\left\{{f(\lambda)g(\lambda)\over \lambda}: {\lambda\in\sigma(A)}\right\}$ and $k'=\max\left\{{f(\lambda)g(\lambda)\over \lambda}: {\lambda\in\sigma(A)\cup\sigma(B)}\right\}$.

Some Berezin number inequalities for operator matrices

Abstract: The Berezin symbol $\widetilde{A}$ of an operator $A$ acting on the reproducing kernel Hilbert space ${\mathscr H}={\mathscr H(}\Omega)$ over some (non-empty) set is defined by $\widetilde{A}(\lambda)=\langle A\hat{k}_{\lambda},\hat{k}_{\lambda}\rangle\,\,\,(\lambda\in\Omega)$, where $\hat{k}_{\lambda}=\frac{{k}_{\lambda}}{\|{k}_{\lambda}\|}$ is the normalized reproducing kernel of ${\mathscr H}$. The Berezin number of operator $A$ is defined by $\mathbf{ber}(A) = \underset{\lambda \in \Omega}{\sup} \big|\tilde{A}(\lambda)\big|=\underset{\lambda \in \Omega }{\sup} \big|\langle A\hat{k}_{\lambda}, \hat{k}_{\lambda}\rangle\big|$. 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 $\mathbf{T}=\left[\begin{array}{cc} A&B C&D \end{array}\right]\in {\mathbb B}({\mathscr H(\Omega_1)}\oplus{\mathscr H(\Omega_2)})$, 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*}

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

Abstract: Motivated by some recently established Jensen-type operator inequalities related to a convex function, in the present paper we derive several more accurate Jensen-type operator inequalities for certain subclasses of convex functions. More precisely, we obtain interpolating series of Jensen-type inequalities utilizing 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.

Improvements of Berezin number inequalities.

Abstract: 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_{\alpha}(A,B))&\leq\frac{\|X\|^{r}}{2}\textbf{ber}(A^{r}+B^{r})&\leq\frac{\|X\|^{r}}{2}\textbf{ber}(\alpha A^{r}+(1-\alpha)B^{r})+\textbf{ber}((1-\alpha)A^{r}+\alpha B^{r}), \end{align*} where $H_{\alpha}(A,B)=\frac{A^\alpha XB^{1-\alpha}+A^{1-\alpha} XB^{\alpha}}{2}$, $0\leq\alpha\leq1$ and $r\geq2$.

Some extensions of the Young and Heinz inequalities for matrices

Abstract: 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 $$\begin{aligned}&\Big \Vert A^{ u }XB^{1- u }+A^{1- u }XB^{ u }\Big \Vert _{2}^{2}\le \Big \Vert AX+XB\Big \Vert _{2}^{2}- 2r\Big \Vert AX-XB\Big \Vert _{2}^{2}\\&\quad -r_{0}\left( \Big \Vert A^{\frac{1}{2}}XB^{\frac{1}{2}}-AX\Big \Vert _{2}^{2}+ \Big \Vert A^{\frac{1}{2}}XB^{\frac{1}{2}}-XB\Big \Vert _{2}^{2}\right) , \end{aligned}$$ where X is an arbitrary $$n\times n$$ matrix, $$0< u \le \frac{1}{2}$$ , $$r=\min \{ u , 1- u \}$$ and $$r_{0}=\min \{2r, 1-2r\}$$ .

Generalizations of the Aluthge transform of operators

Abstract: 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 $\\tilde{A}_{f,g} =f(|A|) U g(|A|)$.

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

Abstract: 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

Further refinements of generalized numerical radius inequalities for Hilbert space operators

Abstract: 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}\eta(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*} \eta(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*}

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

Abstract: 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.

Some generalized numerical radius inequalities involving Kwong functions

Abstract: 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 \[ \omega(H_{f,g}(A))\leq k\, \omega(AX+XA), \] which is equivalent to \[\omega\big(H_{f,g}(A,B)\pm H_{f,g}(B,A)\big)\leq k'\,\left\{\omega((A+B)X+X(A+B))+\omega((A-B)X-X(A-B))\right\},\] where $f$ and $g$ are two continuous functions on $(0,\infty)$ such that $h(t)={f(t)\over g(t)}$ is Kwong, $k=\max\left\{{f(\lambda)g(\lambda)\over \lambda}: {\lambda\in\sigma(A)}\right\}$ and $k'=\max\left\{{f(\lambda)g(\lambda)\over \lambda}: {\lambda\in\sigma(A)\cup\sigma(B)}\right\}$.

Berezin number inequalities for Hilbert space operators

Abstract: 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 $$\mathbf{ber}^2(A^*XB)\leqslant\|X\|^2\mathbf{ber}(A^*A)\mathbf{ber}(B^*B).$$

Unitarily invariant norm inequalities involving $G_1$ operators

Abstract: 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 $\sigma (A)\cup\sigma(B)\subset\mathbb{D}$ and $f, g$ are analytic on the complex unit disk $\mathbb{{D}}$, $g(0)=f(0)=1$, $\textrm{Re}(f)>0$ and $\textrm{Re}(g)>0$.

Some matrix power and Karcher means inequalities involving positive linear maps

Abstract: In this paper, we generalize some matrix inequalities involving the matrix power means and Karcher mean of positive definite matrices. Among other inequalities, it is shown that if A = (A1,...,An) is an n-tuple of positive definite matrices such that 0 < m ≤ Ai ≤ M (i = 1,...,n) for some scalars m < M and ω = (w1,...,wn) is a weight vector with wi ≥ 0 and Σn,i=1 wi=1, then Фp (Σn,i=1 wiAi)≤ αpфp(Pt(ω,A)) and фp (Σn,i=1 wiAi) ≤ αpфp(Λ(ω,A)), where p > 0,α = max {(M+m)2/4Mm,(M+m)2/42p Mm}, Ф is a positive unital linear map and t  [-1,1]\{0}.

Reverses of Ando’s and Hölder–McCarty’s inequalities

Abstract: In this paper, we give some reverse-types of Ando’s and Holder–McCarthy’s inequalities for positive linear maps, and positive invertible operators. For this purpose, we use a recently improved Young inequality and its reverse.

Reverses of Ando's and H\"older-Macarty's inequalities

Abstract: In this paper, we give some reverse-types of Ando's and Holder-McCarthy's inequalities for positive linear maps, and positive invertible operators. For our purpose, we use a recently improved Young inequality and its reverse.

Noncommutative Chebyshev inequality involving the Hadamard product

Abstract: 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 $\mu$ as a totaly order set, then \begin{align*} \int_{T} \alpha(s) d\mu(s)\int_{T}\alpha(t)(A_t\circ B_t) d\mu(t)\geq\Big{(}\int_{T}\alpha(t) (A_tm_{r,\alpha} B_t) d\mu(t)\Big{)}\circ\Big{(}\int_{T}\alpha(s) (A_sm_{r,1-\alpha} B_s) d\mu(s)\Big{)}, \end{align*} where $\alpha\in[0,1]$, $r\in[-1,1]$ and $(A_t)_{t\in T}, (B_t)_{t\in T} $ are positive increasing fields in $\mathcal{C}(T,\mathfrak A)$.