Journal of Mathematical Analysis and Applications, ( ISI ), Volume (445), No (2), Year (2017-1) , Pages (1516-1529)

Title : ( Inequalities for generalized Euclidean operator radius via Young's inequality )

Authors: Alemeh Sheikhhosseini , Mohammad Sal Moslehian , Khalid Shebrawi ,

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Abstract

Using a refinement of the classical Young inequality, we refine some inequalities of operators including the function $\omega_{p}$, where $% \omega_{p}$ is defined for $p \geqslant 1$ and operators $T_{1}, \ldots, T_{n} \in \mathbb{B}(\mathscr{H})$ by \begin{equation*} \omega_{p}(T_{1}, \ldots, T_{n}):=\sup_{\|x\|=1} \left( \sum_{i=1}^{n} | \left \langle T_{i}x, x \right\rangle|^{p} \right) ^{\frac{1}{p}}. \end{equation*} Among other things, we show that if $T_{1}, \ldots, T_{n} \in \mathbb{B}(% \mathscr{H})$ and $p \geq q \geq 1$ with $\frac{1}{p}+\frac{1}{q} =1,$ then \begin{align*} \frac{1}{n}\left \| \sum_{i=1}^{n}T_{i} \right \|^{2} & \leq \omega_{p}(|T_{1}|, \ldots, |T_{n}|)\omega_{q}(|T_{1}^{*}|, \ldots, |T_{n}^{*}|) \\ & \leq \frac{1}{p} \left\| \sum_{i=1}^{n} | T_{i}|^{p}\right \| + \frac{1}{q} \left\| \sum_{i=1}^{n} | T_{i}^{*}|^{q}\right\| -\inf_{\|x\|=\|y\|=1} \delta(x, y), \end{align*}% where $\delta(x, y) = \frac{1}{p} \left( \sqrt{\sum_{i=1}^{n} \left \langle |T_{i}|x, x \right\rangle^{p} } - \sqrt{\sum_{i=1}^{n} \left \langle |T_{i}^{*}|y, y \right\rangle^{q}} \right)^{2}. $

Keywords

Young inequality; generalized Euclidean operator radius; inequality; numerical range
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@article{paperid:1057236,
author = {Alemeh Sheikhhosseini and Sal Moslehian, Mohammad and Khalid Shebrawi},
title = {Inequalities for generalized Euclidean operator radius via Young's inequality},
journal = {Journal of Mathematical Analysis and Applications},
year = {2017},
volume = {445},
number = {2},
month = {January},
issn = {0022-247X},
pages = {1516--1529},
numpages = {13},
keywords = {Young inequality; generalized Euclidean operator radius; inequality; numerical range},
}

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%0 Journal Article
%T Inequalities for generalized Euclidean operator radius via Young's inequality
%A Alemeh Sheikhhosseini
%A Sal Moslehian, Mohammad
%A Khalid Shebrawi
%J Journal of Mathematical Analysis and Applications
%@ 0022-247X
%D 2017

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