Subnormality of 2-variable weighted shifts with diagonal core

Raúl Enrique Curto; Sang Hoon Lee; Jasang Yoon

doi:10.1016/j.crma.2013.03.002

Functional Analysis

Subnormality of 2-variable weighted shifts with diagonal core

Presented by: Gilles Pisier

Raúl Enrique Curto ¹; Sang Hoon Lee ²; Jasang Yoon ³

¹ Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA
² Department of Mathematics, Chungnam National University, Daejeon, 305-764, Republic of Korea
³ Department of Mathematics, The University of Texas-Pan American, Edinburg, TX 78539, USA

Comptes Rendus. Mathématique, Volume 351 (2013) no. 5-6, pp. 203-207.

Abstract
Résumé

The Lifting Problem for Commuting Subnormals (LPCS) asks for necessary and sufficient conditions for a pair of subnormal operators on Hilbert space to admit commuting normal extensions. Given a 2-variable weighted shift T with diagonal core, we prove that LPCS is soluble for T if and only if LPCS is soluble for some power $T^{m} (m \in Z_{+}^{2}, m \equiv (m_{1}, m_{2}), m_{1}, m_{2} ⩾ 1)$ . We do this by first developing the basic properties of diagonal cores, and then analyzing how a diagonal core interacts with the rest of the 2-variable weighted shift.

Le problème du relèvement des opérateurs sous-normaux commutatifs (LPCS) consiste à rechercher des conditions nécessaires ou suffisantes pour que deux opérateurs sous-normaux sur lʼespace de Hilbert admettent des extensions normales commutatives. Étant donné un opérateur de décalage pondéré T à deux variables avec cœur diagonal, nous prouvons que le LPCS est résoluble pour T si et seulement si le LPCS est résoluble pour une certaine puissance $T^{m} (m \in Z_{+}^{2}, m \equiv (m_{1}, m_{2}), m_{1}, m_{2} ⩾ 1)$ . Nous le faisons en développant dʼabord les propriétés de base des cœurs diagonaux, puis en analysant la façon dont un cœur diagonal interagit avec le reste de lʼopérateur.

Received: 2012-12-28
Accepted: 2013-03-07
Published online: 2013-03-01

DOI: 10.1016/j.crma.2013.03.002

Author's affiliations:

Raúl Enrique Curto ¹; Sang Hoon Lee ²; Jasang Yoon ³

¹ Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA
² Department of Mathematics, Chungnam National University, Daejeon, 305-764, Republic of Korea
³ Department of Mathematics, The University of Texas-Pan American, Edinburg, TX 78539, USA

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Raúl Enrique Curto; Sang Hoon Lee; Jasang Yoon. Subnormality of 2-variable weighted shifts with diagonal core. Comptes Rendus. Mathématique, Volume 351 (2013) no. 5-6, pp. 203-207. doi : 10.1016/j.crma.2013.03.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.03.002/

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Cited by Sources:

^☆ The first named author was partially supported by NSF Grants DMS-0400741 and DMS-0801168. The second named author was partially supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2009-0085279). The third named author was partially supported by a Faculty Research Council Grant at The University of Texas-Pan American.

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