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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

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.

Abstract

Let T be a contraction and A the strong limit of {TnTn}n1. 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.