Let be a commuting n-tuple of operators on a Hilbert space , and let be its canonical joint polar decomposition (i.e. , a joint partial isometry, and ). The spherical Aluthge transform of T is the (necessarily commuting) n-tuple . We prove that , where 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 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.
Soit un n-uplet commutatif d'opérateurs sur un espace de Hilbert , et soient sa décomposition polaire jointe canonique (i.e. , une isométrie partielle jointe et ). La transformée d'Aluthge sphérique de T est le n-uplet (nécessairement commutatif) . Nous démontrons que , où 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 implique l'inversibilité de P. Comme conséquence, nous pouvons étendre notre résultat à d'autres systèmes spectraux définis à partir des complexes de Koszul.
Accepted:
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Chafiq Benhida 1; Raúl E. Curto 2; Sang Hoon Lee 3; Jasang Yoon 4
@article{CRMATH_2019__357_10_799_0, 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}, language = {en}, }
TY - JOUR AU - Chafiq Benhida AU - Raúl E. Curto AU - Sang Hoon Lee AU - Jasang Yoon TI - Joint spectra of spherical Aluthge transforms of commuting n-tuples of Hilbert space operators JO - Comptes Rendus. Mathématique PY - 2019 SP - 799 EP - 802 VL - 357 IS - 10 PB - Elsevier DO - 10.1016/j.crma.2019.10.003 LA - en ID - CRMATH_2019__357_10_799_0 ER -
%0 Journal Article %A Chafiq Benhida %A Raúl E. Curto %A Sang Hoon Lee %A Jasang Yoon %T Joint spectra of spherical Aluthge transforms of commuting n-tuples of Hilbert space operators %J Comptes Rendus. Mathématique %D 2019 %P 799-802 %V 357 %N 10 %I Elsevier %R 10.1016/j.crma.2019.10.003 %G en %F CRMATH_2019__357_10_799_0
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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