Document Type

Article

Publication Date

2020

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:=T∗1T1+⋯+T∗dTd−−−−−−−−−−−−−−−√, and let ⎛⎝⎜⎜T1⋮Td⎞⎠⎟⎟=⎛⎝⎜⎜V1⋮Vd⎞⎠⎟⎟P be the canonical polar decomposition, with (V1,⋯,Vd) a (joint) partial isometry and ⋂i=1dkerTi=⋂i=1dkerVi=kerP. \medskip For 0≤t≤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≤t≤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 Δ(n)t denotes the n--th iterate of Δt. \ For d=t=1, we give an example where the above formula fails.

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