The Spectral Picture and Joint Spectral Radius of the Generalized Spherical Aluthge Transform

TL;DR

This paper fully characterizes the spectral properties of the generalized spherical Aluthge transform for commuting operator tuples and establishes a method to compute the joint spectral radius from the transform's iterates.

Contribution

It provides a comprehensive spectral analysis of the generalized spherical Aluthge transform and links the spectral radius to the iterates of this transform, extending previous results.

Findings

Same spectra for the transform and original tuple

Spectral radius can be obtained from iterates of the transform

Counterexample where the spectral radius formula fails

Abstract

For an arbitrary commuting $d$--tuple $\bT$ of Hilbert space operators, we fully determine the spectral picture of the generalized spherical Aluthge transform $\dbT$ and we prove that the spectral radius of $\bT$ can be calculated from the norms of the iterates of $\dbT$. \ Let $\bm{T} \equiv (T_1,\cdots,T_d)$ be a commuting $d$--tuple of bounded operators acting on an infinite dimensional separable Hilbert space, let $P:=\sqrt{T_1^*T_1+\cdots+T_d^*T_d}$, and let $$ \left( \begin{array}{c} T_1 \\ \vdots \\ T_d \end{array} \right) = \left( \begin{array}{c} V_1 \\ \vdots \\ V_d \end{array} \right) P $$ be the canonical polar decomposition, with $(V_1,\cdots,V_d)$ a (joint) partial isometry and $$ \bigcap_{i=1}^d \ker T_i=\bigcap_{i=1}^d \ker V_i=\ker P. $$ \medskip For $0 \le t \le 1$, we define the generalized spherical Aluthge transform of $\bm{T}$ by $$ \Delta_t(\bm{T}):=(P^t V_1P^{1-t}, \cdots, P^t V_dP^{1-t}). $$ We also let $\left\|\bm{T}\right\|_2:=\left\|P\right\|$. \ We first determine the spectral picture of $\Delta_t(\bm{T})$ in terms of the spectral picture of $\bm{T}$; in particular, we prove that, for any $0 \le t \le 1$, $\Delta_t(\bm{T})$ and $\bm{T}$ 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 $r(\bm{T})$, and prove that $r(\bm{T})=\lim_n\left\|\Delta_t^{(n)}(\bm{T})\right\|_2 \,\, (0 < t < 1)$, where $\Delta_t^{(n)}$ denotes the $n$--th iterate of $\Delta_t$. \ For $d=t=1$, we give an example where the above formula fails.