  Let $\mathbf{T}=(T_1,\ldots,T_d)$ 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
$\widehat{\mathbf{T}}:=(\sqrt{P}V_1\sqrt{P},\ldots,\sqrt{P}V_d\sqrt{P})$ where
$P:=\sqrt{T_1^*T_1+\ldots+T_d^*T_d}$ and $(V_1,\ldots,V_d)$ 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 $\widehat{\mathbf{T}}$. Moreover, a characterization of the
joint spectral radius of an operator tuple $\mathbf{T}$ via $n$-th iterated of
spherical Aluthge transform is established.
