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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Incompatibility of compact perturbations with the Sz. Nagy-Foias functional calculus


Authors: Kenneth R. Davidson and Fouad Zarouf
Journal: Proc. Amer. Math. Soc. 121 (1994), 519-522
MSC: Primary 47A60; Secondary 47D25
DOI: https://doi.org/10.1090/S0002-9939-1994-1195716-3
MathSciNet review: 1195716
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Abstract: For every absolutely continuous contraction T with spectrum on the unit circle, we exhibit an $ {H^\infty }$ function h and a sequence of operators $ {T_n}$ which are unitarily equivalent to T and differ from T by a sequence of compact operators converging to 0 in norm such that $ h({T_n})$ is never a compact perturbation of $ h(T)$. When T is diagonal, it can also be arranged that $ T - {T_n}$ is trace class, and $ {T_n}$ commutes with T.

Pour toute contraction absolument continue T dont le spectre rencontre le cercle unité, il existe une fonction h de $ {H^\infty }$ et une suite $ {T_n}$ d'opérateurs unitairement equivalents à T telle que $ T - {T_n}$ soit compact et convergent en norme vers 0, mais $ h({T_n}) - h(T)$ soit non compact pour tout n. Dans le cas où T est diagonal, la suite $ {T_n}$ vérifie en plus $ T - {T_n}$ est un opérateur à trace, et $ {T_n}$ commute avec T.


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DOI: https://doi.org/10.1090/S0002-9939-1994-1195716-3
Article copyright: © Copyright 1994 American Mathematical Society