Hyperinvariant subspaces of 𝐶₁₁ contractions

Abstract

For an operator T on a Hilbert space let Hyperlat T denote its hyperinvariant subspace lattice. Assume that T is a completely nonunitary C 11 {C_{11}} contraction with finite defect indices. In this note we characterize the elements of Hyperlat T among invariant subspaces for T in terms of their corresponding regular factorizations and show that elements in Hyperlat T are exactly the spectral subspaces of T defined by Sz.-Nagy and Foiaş. As a corollary, if T 1 , T 2 {T_1},{T_2} are two such operators which are quasi-similar to each other, then Hyperlat T 1 {T_1} is (lattice) isomorphic to Hyperlat T 2 {T_2} .