Functional Analysis
Spectral pictures of 2-variable weighted shifts
Comptes Rendus. Mathématique, Volume 343 (2006) no. 9, pp. 579-584.

We study the spectral pictures of (jointly) hyponormal 2-variable weighted shifts with commuting subnormal components. By contrast with all known results in the theory of subnormal single and 2-variable weighted shifts, we show that the Taylor essential spectrum can be disconnected. We do this by obtaining a simple sufficient condition that guarantees disconnectedness, based on the norms of the horizontal slices of the shift. We also show that for every k1 there exists a k-hyponormal 2-variable weighted shift whose horizontal and vertical slices have 1- or 2-atomic Berger measures, and whose Taylor essential spectrum is disconnected.

Nous étudions les images spectrales de shifts pondérés à deux variables et (conjointement) hyponormaux possédant des composants commutants sousnormaux. À la différence de tous les résultats connus dans la théorie des shifts pondérés simples sousnormaux à deux variables, nous démontrons que le spectre essentiel de Taylor peut être déconnecté. Nous faisons cela en obtenant une condition suffisante simple qui garantit le caractère déconnecté de ce spectre, basée sur les normes des sections horizontales du shift. Nous montrons également que pour chaque k1 il existe un shift pondéré k-hyponormal à deux variables dont le spectre essentiel de Taylor est déconnecté.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2006.09.024
Curto, Raúl E. 1; Yoon, Jasang 2

1 Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242, USA
2 Department of Mathematics, Iowa State University, Ames, Iowa 50011, USA
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Curto, Raúl E.; Yoon, Jasang. Spectral pictures of 2-variable weighted shifts. Comptes Rendus. Mathématique, Volume 343 (2006) no. 9, pp. 579-584. doi : 10.1016/j.crma.2006.09.024. http://www.numdam.org/articles/10.1016/j.crma.2006.09.024/

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Research partially supported by NSF Grants DMS-0099357 and DMS-0400741.