The Spectral picture and joint spectral radius of the generalized spherical Aluthge transform

For an arbitrary commuting d–tuple $\bf T$ of Hilbert space operators, we fully determine the spectral picture of the generalized spherical Aluthge transform $\Delta_t({\bf T}) $ and we prove that the spectral radius of $\bf T$ can be calculated from the norms of the iterates of $\Delta_t({\bf T}) $. We first determine the spectral picture of  $\Delta_t({\bf T}) $  in terms of the spectral picture of $\bf T$; in particular, we prove that, for any $0\leq t\leq 1$,   $\Delta_t({\bf T}) $ and $\bf T$ have the same Taylor spectrum, the same Taylor essential spectrum, the same Fredholm index, and the same Harte spectrum. We then study the joint spectral radius $r({\bf T})$, and prove that  $r({\bf T})=\lim_n\|\Delta_t({\bf T})^{(n)}\|_2$  $  (0<t<1)$, where $\Delta_t({\bf T})^{(n)} $ denotes the n–th iterate of $\Delta_t({\bf T}) $: For d=t=1, we give an example where the above formula fails.

The talk is based on recent research with Chafiq Benhida, Sang Hoon Leeand Jasang Yoon