Quasiaffine transforms of Hilbert space operators

Ampliation quasisimilarity was applied as a tool in Foias and Pearcy (J Funct Anal 219:134-142, 2005) to reduce the hyperinvariant subspace problem to a particular class of operators. The seemingly weaker pluquasisimilarity relation was introduced in Bercovici et al. (Acta Sci Math Szeged 85:681-691, 2019) and studied also in Kerchy (Acta Sci Math Szeged 86:503-520, 2020). The problem whether these two relations are actually equivalent is addressed in the present paper. The following more general, related question is studied in details: under what conditions is the operator A a quasiaffine transform of B, whenever A can be injected into B and A can be also densely mapped into B.