Extended Joint Numerical Radius of the Spherical Aluthge Transform

Let \(T=(T_{1}, T_{2},\ldots , T_{n})\) $T=(T_1,T_2,\dots ,T_n)$ be a commuting \(n-\) $n-$ tuple of operators on a complex Hilbert space H. We define the extended joint numerical radius of T by \(\begin{aligned} J_{t}w_{(N, v)}(T)=\sup \limits _{(\lambda _{1}, \lambda _{2}, \ldots , \lambda _{n})\in \overline{B_{n}}(0, 1)}w_{(N, v)}\bigg (\sum \limits _{i=1}^{n}\lambda _{i}T_{i}\bigg ), \end{aligned}\) $$\begin{array}{c}\hfill J_t{w}_{(N,v)}(T)=\underset{({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n)\in \overline{B_n}(0,1)}{sup}{w}_{(N,v)}(\sum _{i=1}^n{\lambda }_i{T}_i),\end{array}$$ where N is any norm on B(H), \(w_{(N, v)}(S)=\sup \limits _{\theta \in \mathbb {R}}N(ve^{i\theta }S+(1-v)e^{-i\theta }S^{*}), S\in B(H), v\in [0, 1],\) $$w_{(N,v)}(S)=\underset{\theta \in \mathbb{R}}{sup}N(v{e}^{i\theta }S+(1-v)e^{-i\theta }S_{}^{\ast }),S\in B(H),v\in [0,1],$$ and \(\overline{B_{n}}(0, 1)\) $\overline{B_n}(0,1)$ denotes the closure of the unit ball in \(\mathbb {C}^{n}\) ${\mathbb{C}}^n$ with respect to the euclidean norm, i.e. \(\overline{B_{n}}(0, 1)=\left\{ \lambda =(\lambda _{1}, \ldots , \lambda _{n})\in \mathbb {C}^{n}; \parallel \lambda \parallel _{2}=\bigg (\sum \limits _{i=1}^{n}|\lambda _{i}|^{2}\bigg )^{\frac{1}{2}}\le 1 \right\} .\) $$\overline{B_n}(0,1)=\left\{\lambda ,=,({\lambda }_1,\dots ,{\lambda }_n),\in ,{\mathbb{C}}^n,;,\Vert ,\lambda ,{\Vert }_2,=,(,\sum _{i=1}^n,{|{\lambda }_i|}^2,{)}^{\frac{1}{2}},\le ,1\right\}.$$ In this paper, we prove several inequalities for the extended joint numerical radius involving the spherical Aluthge transform in the case where N is the operator norm of B(H) or the numerical radius.