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Bulletin of the London Mathematical Society Advance Access published online on November 1, 2009
Bulletin of the London Mathematical Society, doi:10.1112/blms/bdp084
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© 2009 London Mathematical Society

Multiplicity of direct sums of operators on Banach spaces

Sophie Grivaux

Laboratoire Paul Painlevé, UMR 8524
Université Lille 1, Cité Scientifique
59655 Villeneuve d'Ascq Cedex
France

Maria Roginskaya

Chalmers University of Technology
SE-41296 Göteborg
Sweden
Department of Mathematical Sciences
Göteborg University
SE-41296 Göteborg
Sweden
maria@chalmers.se

Received 17 January 2008. Revision received 1 April 2009.

Let T be a bounded operator on a complex Banach space X and let Tn be the direct sum T{oplus}...{oplus} T of n copies of T acting on X{oplus}...{oplus} X. The aim of this paper is to study the sequence (m(Tn))n ≥ 1 of the multiplicities of the operators Tn. Answering a question of Atzmon, it is shown that this sequence is either eventually constant or grows to infinity at least as fast as n. Then examples of operators on Hilbert spaces, such that m(Tn) = d for every n ≥ 1, are constructed, where d is an arbitrary positive integer. This answers a question of Herrero and Wogen and characterizes convex sequences that can be realized as a sequence (m(Tn))n ≥ 0 for some operator T on a Hilbert space.

2000 Mathematics Subject Classification 47A16, 47B37.

Research supported in part by the ANR Projet Blanc DYNOP.


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