Extended Joint Numerical Radius of the Spherical Aluthge Transform

Let \(T=(T_{1}, T_{2},\ldots , T_{n})\) be a commuting \(n-\)tuple of operators on a complex Hilbert space H. We define the extended joint numerical radius of T by

$$\begin{aligned} J_{t}w_{(N, v)}(T)=\sup \limits _{(\lambda _{1}, \lambda _{2}, \ldots , \lambda _{n})\in \overline{B_{n}}(0, 1)}w_{(N, v)}\bigg (\sum \limits _{i=1}^{n}\lambda _{i}T_{i}\bigg ), \end{aligned}$$

where N is any norm on B(H),

$$w_{(N, v)}(S)=\sup \limits _{\theta \in \mathbb {R}}N(ve^{i\theta }S+(1-v)e^{-i\theta }S^{*}), S\in B(H), v\in [0, 1],$$

and \(\overline{B_{n}}(0, 1)\) denotes the closure of the unit ball in \(\mathbb {C}^{n}\) with respect to the euclidean norm, i.e.

$$\overline{B_{n}}(0, 1)=\left\{ \lambda =(\lambda _{1}, \ldots , \lambda _{n})\in \mathbb {C}^{n}; \parallel \lambda \parallel _{2}=\bigg (\sum \limits _{i=1}^{n}|\lambda _{i}|^{2}\bigg )^{\frac{1}{2}}\le 1 \right\} .$$

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.