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

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 $\mathbf{T}\equiv (T_1,\cdots ,T_d)$ be a commuting $d$--tuple of bounded operators acting on an infinite dimensional separable Hilbert space, let $P:=\sqrt{T_1^{\ast }{T}_1+\cdots +T_d^{\ast }{T}_d}$, and let $$\left(\begin{array}{c}{T}_1\\ ⋮\\ T_d\end{array}\right)=\left(\begin{array}{c}{V}_1\\ ⋮\\ V_d\end{array}\right)P$$ 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 $\mathbf{T}$ by $${\Delta }_t(\mathbf{T}):=(P^t{V}_1{P}^{1-t},\cdots ,P^t{V}_d{P}^{1-t}).$$ We also let ${\Vert \mathbf{T}\Vert }_2:=\Vert P\Vert$. \ We first determine the spectral picture of ${\Delta }_t(\mathbf{T})$ in terms of the spectral picture of $\mathbf{T}$; in particular, we prove that, for any $0\le t\le 1$, ${\Delta }_t(\mathbf{T})$ and $\mathbf{T}$ 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 $r(\mathbf{T})$, and prove that $r(\mathbf{T})={\lim }_n{\Vert {\Delta }_t^{(n)}(\mathbf{T})\Vert }_2(0<t<1)$, 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.