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

Authors

Chafiq Benhida
Raul E. Curto
Sang Hoon Lee
Jasang Yoon, The University of Texas Rio Grande Valley

Document Type

Article

Publication Date

10-29-2022

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.

Comments

Original published version available at https://doi.org/10.1016/j.aim.2022.108602

Recommended Citation

Benhida, Chafiq, et al. "The spectral picture and joint spectral radius of the generalized spherical Aluthge transform." Advances in Mathematics 408 (2022): 108602. https://doi.org/10.1016/j.aim.2022.108602

Publication Title

Advances in Mathematics

DOI

10.1016/j.aim.2022.108602