K. Shebrawi | Searcharxiv

arXiv2017

Sharpening Some Classical Numerical Radius Inequalities

New upper and lower bounds for the numerical radii of Hilbert space operators are given. Among our results, we prove that if $A\in \mathcal{B} \left( \mathcal{H}\right) $ is a hyponormal operator, then for all non-negative non-decreasing operator convex $f$ on $ [0,\infty ),$ we have \[f\left( \omega \left( A \right) \right)\le \frac{1}{2}\left\| f\left( \frac{1}{1+\frac{\xi_{\left| A \right|}^{2}}{8}}\left| A \right| \right)+f\left( \frac{1}{1+\frac{\xi_{\left| A \right|}^{2}}{8}}\left| {{A}^{*}} \right| \right) \right\|,\] where ${{\xi }_{\left| A\right| }}=\underset{\left| x\right| =1}{\mathop{\inf }}\,\left\{ \frac{\left\langle \left( \left| A\right| -\left| {{A}^{\ast }}\right| \right) x,x\right\rangle }{ \left\langle \left( \left| A\right| +\left| {A^{\ast }} \right| \right) x,x\right\rangle }\right\} $. Our results refine and generalize earlier inequalities for hyponormal operator.

math.FA↗

arXiv2015

Extension of Euclidean operator radius inequalities

To extend the Euclidean operator radius, we define $w_p$ for an $n$-tuples of operators $(T_1,\ldots, T_n)$ in $\mathbb{B}(\mathscr{H})$ by $w_p(T_1,\ldots,T_n):= \sup_{\| x \| =1} \left(\sum_{i=1}^{n}| \langle T_i x, x \rangle |^p \right)^{\frac1p}$ for $p\geq1$. We generalize some inequalities including Euclidean operator radius of two operators to those involving $w_p$. Further we obtain some lower and upper bounds for $w_p$. Our main result states that if $f$ and $g$ are nonnegative continuous functions on $\left[ 0,\infty \right) $ satisfying $f\left( t\right) g\left(t\right) =t$ for all $t\in \left[ 0,\infty \right) $, then \begin{equation*} w_{p}^{rp}\left( A_{1}^{\ast }T_{1}B_{1},\ldots ,A_{n}^{\ast }T_{n}B_{n}\right) \leq \frac{1}{2}\left\Vert \underset{i=1}{\overset{n}{\sum }}\Big( \left[ B_{i}^{\ast }f^{2}\left( \left\vert T_{i}\right\vert \right) B_{i}\right] ^{rp}+\left[ A_{i}^{\ast }g^{2}\left( \left\vert T_{i}^{\ast }\right\vert \right) A_{i}\right] ^{rp}\Big)\right\Vert \end{equation*} for all $p\geq 1$, $r\geq 1$ and operators in $ \mathbb{B}(\mathscr{H})$.

math.FA↗