Extension of Euclidean operator radius inequalities

Abstract

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})$.