Joint spectra of spherical Aluthge transforms of commuting <i>n</i>-tuples of Hilbert space operators

Chafiq Benhida; Raúl E. Curto; Sang Hoon Lee; Jasang Yoon

doi:10.1016/j.crma.2019.10.003

Functional analysis

Joint spectra of spherical Aluthge transforms of commuting n-tuples of Hilbert space operators
[Spectres joints des transformées d'Aluthge sphériques de n-uplets commutatifs d'opérateurs d'un espace de Hilbert]

Présenté par : Gilles Pisier

Chafiq Benhida ¹ ; Raúl E. Curto ² ; Sang Hoon Lee ³ ; Jasang Yoon ⁴

¹ UFR de mathématiques, Université des sciences et technologies de Lille, 59655 Villeneuve-d'Ascq cedex, France
² Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA
³ Department of Mathematics, Chungnam National University, Daejeon, 34134, Republic of Korea
⁴ School of Mathematical and Statistical Sciences, The University of Texas Rio Grande Valley, Edinburg, TX 78539, USA

Comptes Rendus. Mathématique, Volume 357 (2019) no. 10, pp. 799-802.

Résumé
Abstract

Soit $T \equiv (T_{1}, \dots, T_{n})$ un n-uplet commutatif d'opérateurs sur un espace de Hilbert $H$ , et soient $T_{i} \equiv V_{i} P (1 \leq i \leq n)$ sa décomposition polaire jointe canonique (i.e. $P : = \sqrt{T_{1}^{⁎} T_{1} + \dots + T_{n}^{⁎} T_{n}}$ , $(V_{1}, \dots, V_{n})$ une isométrie partielle jointe et $⋂_{i = 1}^{n} \ker T_{i} = ⋂_{i = 1}^{n} \ker V_{i} = \ker P$ ). La transformée d'Aluthge sphérique de T est le n-uplet (nécessairement commutatif) $\hat{T} : = (\sqrt{P} V_{1} \sqrt{P}, \dots, \sqrt{P} V_{n} \sqrt{P})$ . Nous démontrons que $σ_{T} (\hat{T}) = σ_{T} (T)$ , où $σ_{T}$ désigne le spectre de Taylor. Nous procédons pour cela en deux étapes : en dehors de l'origine, nous utilisons les outils et les techniques de la commutativité criss-cross ; à l'origine, nous prouvons que l'inversibilité à gauche de T ou de $\hat{T}$ implique l'inversibilité de P. Comme conséquence, nous pouvons étendre notre résultat à d'autres systèmes spectraux définis à partir des complexes de Koszul.

Let $T \equiv (T_{1}, \dots, T_{n})$ be a commuting n-tuple of operators on a Hilbert space $H$ , and let $T_{i} \equiv V_{i} P (1 \leq i \leq n)$ be its canonical joint polar decomposition (i.e. $P : = \sqrt{T_{1}^{⁎} T_{1} + \dots + T_{n}^{⁎} T_{n}}$ , $(V_{1}, \dots, V_{n})$ a joint partial isometry, and $⋂_{i = 1}^{n} \ker T_{i} = ⋂_{i = 1}^{n} \ker V_{i} = \ker P$ ). The spherical Aluthge transform of T is the (necessarily commuting) n-tuple $\hat{T} : = (\sqrt{P} V_{1} \sqrt{P}, \dots, \sqrt{P} V_{n} \sqrt{P})$ . We prove that $σ_{T} (\hat{T}) = σ_{T} (T)$ , where $σ_{T}$ denotes the Taylor spectrum. We do this in two stages: away from the origin, we use tools and techniques from criss-cross commutativity; at the origin, we show that the left invertibility of T or $\hat{T}$ implies the invertibility of P. As a consequence, we can readily extend our main result to other spectral systems that rely on the Koszul complex for their definitions.

Reçu le : 2019-06-27
Accepté le : 2019-10-07
Publié le : 2019-10-01

DOI : 10.1016/j.crma.2019.10.003

Affiliations des auteurs :

Chafiq Benhida ¹ ; Raúl E. Curto ² ; Sang Hoon Lee ³ ; Jasang Yoon ⁴

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     author = {Chafiq Benhida and Ra\'ul E. Curto and Sang Hoon Lee and Jasang Yoon},
     title = {Joint spectra of spherical {Aluthge} transforms of commuting \protect\emph{n}-tuples of {Hilbert} space operators},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {799--802},
     publisher = {Elsevier},
     volume = {357},
     number = {10},
     year = {2019},
     doi = {10.1016/j.crma.2019.10.003},
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Chafiq Benhida; Raúl E. Curto; Sang Hoon Lee; Jasang Yoon. Joint spectra of spherical Aluthge transforms of commuting n-tuples of Hilbert space operators. Comptes Rendus. Mathématique, Volume 357 (2019) no. 10, pp. 799-802. doi : 10.1016/j.crma.2019.10.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.10.003/

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