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$C_{\cdot 0}$-contractions: a Jordan model and lattices of invariant subspaces


Author: M. F. Gamal'
Translated by: V. V. Kapustin
Original publication: Algebra i Analiz, tom 15 (2003), nomer 5.
Journal: St. Petersburg Math. J. 15 (2004), 773-793
MSC (2000): Primary 47A15, 47A45
DOI: https://doi.org/10.1090/S1061-0022-04-00831-3
Published electronically: July 29, 2004
MathSciNet review: 2068794
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Abstract: A subclass $C_{\cdot 0}(C_0(P), \mathrm{fin})$ of the class of $C_{\cdot 0}$-contractions is introduced and studied. This subclass is a generalization of the subclass of $C_{\cdot 0}$-contractions with finite defect indices, and it includes the $C_{\cdot 0}$-contractions $T$ for which $\operatorname{dim}{ \operatorname{Ker}{T^\ast}}\allowbreak <\infty$ and the defect operator $(I-T^\ast T)^{1/2}$ belongs to the Hilbert-Schmidt class. For an operator of class $C_{\cdot 0}(C_0(P), \mathrm{fin})$, a Jordan model is constructed, and it is proved that the lattices of invariant subspaces remain isomorphic under the quasiaffine transformations.


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Additional Information

M. F. Gamal'
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg, 191023, Russia
Email: gamal@pdmi.ras.ru

DOI: https://doi.org/10.1090/S1061-0022-04-00831-3
Keywords: $C_{\cdot 0}$-contractions, invariant subspaces, quasiaffine transformation, Jordan model, property $(P)$ for $C_0$-contractions
Received by editor(s): March 3, 2003
Published electronically: July 29, 2004
Additional Notes: Partially supported by RFBR (grants nos. 02-01-00264 and NSh-2266.2003.1).
Dedicated: Dedicated to the memory of Professor Yuri A. Abramovich
Article copyright: © Copyright 2004 American Mathematical Society

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