Proper contractions and invariant subspaces (Q1598305)

Summary: Let \(T\) be a contraction and \(A\) the strong limit of \(\{T^{* n}T^n\}_{n\geq 1}\). 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 \({\mathcal C}_{00}\) or a nonstrict proper contraction of class \({\mathcal C}_{10}\) for which \(A\) is a completely nonprojective nonstrict proper contraction. Moreover, its self-commutator \([T^*,T]\) is a strict contraction.