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International Journal of Mathematics and Mathematical Sciences
Volume 28 (2001), Issue 4, Pages 223-230
http://dx.doi.org/10.1155/S0161171201006287

Proper contractions and invariant subspaces

C. S. Kubrusly1 and N. Levan2

1Catholic University of Rio de Janeiro, Rio de Janeiro, RJ 22453-900, Brazil
2University of California at Los Angeles, Los Angeles, CA 90024-1594, USA

Received 4 December 2000

Copyright © 2001 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let T be a contraction and A the strong limit of {T∗nTn}n≥1. We prove the following theorem: if a hyponormal contraction T does not have a nontrivial invariant subspace, then T is either a proper contraction of class 𝒞00 or a nonstrict proper contraction of class 𝒞10 for which A is a completely nonprojective nonstrict proper contraction. Moreover, its self-commutator [T*,T] is a strict contraction.