Extension of Euclidean operator radius inequalities

Abstract

To extend the Euclidean operator radius, we define $w_p$ for an $n$-tuples of operators $(T_1,\dots ,T_n)$ in $\mathbb{B}(\mathcal{H})$ by $w_p(T_1,\dots ,T_n):={\sup }_{\Vert x\Vert =1}{\left({\sum }_{i=1}^n|⟨T_{i}x,x⟩{|}^p\right)}^{\frac{1}{p}}$ for $p\ge 1$. 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 $p\ge 1$, $r\ge 1$ and operators in $\mathbb{B}(\mathcal{H})$.

16 papers, to appear in Mathematica Scandinavica (Math Scand)