Kérchy László honlapja

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Cím:On the hyperinvariant subspace problem for asymptotically nonvanishing contractions.
Szerző:K\'erchy, L\'aszl\'o
Forrás:K\'erchy, L\'aszl\'o (ed.) et al., Recent advances in operator theory and related topics. The B\'ela Sz\H{o}kefalvi-Nagy memorial volume. Proceedings of the memorial conference, Szeged, Hungary, August 2-6, 1999. Basel: Birkh\"auser. Oper. Theory, Adv. Appl. 127, 399-422 (2001). [ISBN 3-7643-6607-9/hbk; ISSN 0378-620X]
Nyelv:English
Absztrakt:The contents of the paper under review are best described by the author's own words: ``Our aim in the present note is to obtain new information on the structure of contractions of class $C_{1,0}$, as well as to obtain new criteria for the existence of hyperinvariant subspaces for these operators. In particular, we identify a new spectral invariant $\pi(T)$, which plays a relevant role in this topic and seems to be a useful tool in the study of $C_{1,0}$-contractions.'' \par To explain what the aforementioned spectral invariant is, consider an absolutely continuous contraction $T$ on some complex Hilbert space ${\Cal H}$. Then $\pi(T)$ is the largest Borel subset $\alpha$ of the unit circle ${\Bbb T}$ having the following property: If $F=\{f_n\}_{n\ge 1}$ is a sequence in $H^\infty$ such that $|f_1(z)|\ge|f_2(z)|\ge\cdots$ for all $z$ in the unit disk ${\Bbb D}$, and $\chi_\alpha\varphi_F\ne 0$, then $\liminf_{n\to\infty}\|f_n(T)x\|>0$ whenever $0\ne x\in{\Cal H}$, where $f_n(T)$ is defined by the Sz.-Nagy--Foia\c{s} functional calculus, $\chi_\alpha$ is the characteristic function of $\alpha$, and $\varphi_F(z):=\lim_{n\to\infty}|f_n(z)|$ for almost every $z\in{\Bbb T}$.} \RV{Daniel Beltita (Bucure\c{s}ti) 
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