Joint numerical radius of spherical Aluthge transforms of tuples of Hilbert space operators

Abstract

Let $\mathbf{T}=(T_1,\dots ,T_d)$ be a $d$-tuple of operators on a complex Hilbert space $\mathcal{H}$. The spherical Aluthge transform of $\mathbf{T}$ is the $d$-tuple given by $\hat{\mathbf{T}}:=(\sqrt{P}{V}_1\sqrt{P},\dots ,\sqrt{P}{V}_d\sqrt{P})$ where $P:=\sqrt{T_1^{\ast }{T}_1+\dots +T_d^{\ast }{T}_d}$ and $(V_1,\dots ,V_d)$ is a joint partial isometry such that $T_k=V_{k}P$ for all $1\le k\le d$. In this paper, we prove several inequalities involving the joint numerical radius and the joint operator norm of $\hat{\mathbf{T}}$. Moreover, a characterization of the joint spectral radius of an operator tuple $\mathbf{T}$ via $n$-th iterated of spherical Aluthge transform is established.