arXiv subjects

Monire Hajmohamadi

Publications and source records attributed to Monire Hajmohamadi.

7 recordsLinked to original sources

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

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↗

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

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 inequalities of matrix power and Karcher means for positive linear maps

In this paper, we generalize some matrix inequalities involving matrix power and Karcher means of positive definite matrices. Among other inequalities, it is shown that if ${\mathbb A}=(A_{1},...,A_{n})$ is a $n$-tuple of positive definite matrices such that $0 0$, $α=\max\Big\{\frac{(M+m)^{2}}{4Mm}, \frac{(M+m)^{2}}{4^{\frac{2}{p}}Mm}\Big\}$, $Φ$ is a positive unital linear map and $t\in [-1, 1]\backslash \{0\}$.

math.FA↗