Generalizations of the relation of quasisimilarity for operators

In this note we first briefly review the progress on the hyperinvari-ant subspace problem for operators on Hilbert space made possible by the equivalence relation of ampliation quasisimilarity recently introduced in [7]. Then we introduce another equivalence relation, which we call pluquasisimilarity , with bigger equivalence classes than ampliation quasisimilarity but very different in appearance, which preserves the existence of hyperinvariant sub-spaces for operators, and thus may be useful in the future. We also compare these with two other equivalence relations, injection-similarity and complete injection-similarity, introduced long ago by Sz.-Nagy and Foias in [13].