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

For an arbitrary commuting $d$–tuple $\text{bT}$ of Hilbert space operators, we fully determine the spectral picture of the generalized spherical Aluthge transform $\text{dbT}$ and we prove that the spectral radius of $\text{bT}$ can be calculated from the norms of the iterates of $\text{dbT}$. \ Let $\mathit{T}\equiv (T_1,\cdots ,T_d)$ 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 $$\left(\begin{array}{c}{T}_1\\ \mathrm{⋮}\\ T_d\end{array}\right)=\left(\begin{array}{c}{V}_1\\ \mathrm{⋮}\\ V_d\end{array}\right)P$$ be the canonical polar decomposition, with $(V_1,\cdots ,V_d)$ a (joint) partial isometry and $$\bigcap _{i=1}^d\ker T_i=\bigcap _{i=1}^d\ker V_i=\ker P\mathrm{.}$$ \medskip For $0\le t\le 1$, we define the generalized spherical Aluthge transform of $\mathit{T}$ by $${\mathrm{\Delta }}_t(\mathit{T}):=(P^t{V}_1{P}^{1-t},\cdots ,P^t{V}_d{P}^{1-t})\mathrm{.}$$ We also let ${\parallel \mathit{T}\parallel }_2:=\parallel P\parallel$. \ We first determine the spectral picture of ${\mathrm{\Delta }}_t(\mathit{T})$ in terms of the spectral picture of $\mathit{T}$; in particular, we prove that, for any $0\le t\le 1$, ${\mathrm{\Delta }}_t(\mathit{T})$ and $\mathit{T}$ 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 $r(\mathit{T})$, and prove that $r(\mathit{T})={\lim }_n{\parallel {\mathrm{\Delta }}_t^{(n)}(\mathit{T})\parallel }_2(0<t<1)$, where ${\mathrm{\Delta }}_t^{(n)}$ denotes the $n$–th iterate of ${\mathrm{\Delta }}_t$. \ For $d=t=1$, we give an example where the above formula fails.