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

Proper contractions and invariant subspaces

C. S. Kubrusly1 and N. Levan2

 1Catholic University of Rio de Janeiro, Rio de Janeiro 22453-900, RJ, Brazil
2University of California at Los Angeles, Los Angeles 90024-1594, CA, USA
Received 4 December 2000

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.