Limits of iterates of spherical Aluthge transforms

Let $\mathbf{T} \equiv (T_1,T_2)$ be a commuting pair of Hilbert space operators, and let $P:=\sqrt{T_1^{\ast}T_1+T_2^{\ast}T_2}$ be the positive factor in the
(joint) polar decomposition of $\mathbf{T}$, i.e., $T_i=V_iP \; (i=1,2)$. The spherical Aluthge transform of $\mathbf{T}$ is the (necessarily commuting) pair
$\widehat{\mathbf{T}}:=(\sqrt{P}V_1\sqrt{P},\sqrt{P}V_2\sqrt{P})$. We study the iterates of the spherical Aluthge transform, that is, the commuting pairs
given by $\widehat{\mathbf{T}}^{(1)}:=\widehat{\mathbf{T}}$ and $\widehat{\mathbf{T}}^{(n)}:=\widehat{\widehat{\mathbf{T}}^{(n-1)}} \; (n \ge 2)$. In this talk, we will focus on the asymptotic behavior of the sequence $\{\widehat{\mathbf{T}}^{(n)}\}_{n \ge 1}$ as $n \rightarrow \infty$. In those cases when the limit exists, the limit pair is a fixed point for the spherical Aluthge transform, that is, a spherically quasinormal pair. For large suitable classes of $2$-variable weighted shifts we will establish the convergence of the sequence of iterates in the weak operator topology.

The talk is based on joint work with Chafiq Benhida (Université de Lille, Lille, France), and with Jasang Yoon (The University of Texas Rio Grande Valley, Edinburg, Texas, USA).