Extensions of Euclidean operator radius inequalities

Abstract

To extend the Euclidean operator radius, we define $w_p$ for an $n$-tuple of operators $(T_1,\dots,T_n)$ in $\mathbb{B}(\mathscr{H})$ by $w_p(T_1,\dots,T_n):= \sup_{\lVert x \rVert =1} (\sum_{i=1}^{n}\lvert \langle T_i x, x \rangle \rvert^p)^{1/p}$ for $p\geq1$. We generalize some inequalities including the 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 non-negative continuous functions on $[0,\infty)$ satisfying $f(t) g(t) =t$ for all $t\in [ 0,\infty)$, then $$\begin{equation*} w_{p}^{rp}( A_{1}^*T_{1}B_{1}, \dots ,A_{n}^*T_{n}B_{n}) \leq \frac{n^{r-1}}{2} \Bigl\lVert \sum_{i=1}^n [ B_{i}^*f^{2}( \lvert T_{i}\rvert ) B_{i}] ^{rp} + [ A_{i}^*g^{2}( \lvert T_{i}^* \rvert ) A_{i}]^{rp} \Bigr\rVert, \end{equation*}$$ for all $p\geq 1$, $r\geq 1$ and operators in $\mathbb{B}(\mathscr{H})$.