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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
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.

References [Enhancements On Off] (What's this?)

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