Abstract. In this note we first briefly review the progress on the hyperinvariant subspace problem for operators on Hilbert space made possible by the equivalence relation of ampliation quasisimilarity recently introduced in \cite {FP}. Then we introduce another equivalence relation, which we call \emph {pluquasisimilarity}, with bigger equivalence classes than ampliation quasisimilarity but very different in appearance, which preserves the existence of hyperinvariant subspaces 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 \cite {SzNF}.

DOI: 10.14232/actasm-019-765-9

AMS Subject Classification (1991): 47A15; 47A65

Keyword(s): hyperinvariant subspace, quasisimilarity, ampliation quasisimilarity, quasiaffinity, quasitriangular operator

Received February 23, 2019 and in final form March 28, 2019. (Registered under 15/2019.)