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 [7].Then we introduce another equivalence relation, which we call pluquasisimilarity, with bigger equivalence classes than ampliation quasisimilarity but verydifferent in appearance, which preserves the existence of hyperinvariant sub-spaces for operators, and thus may be useful in the future. We also comparethese with two other equivalence relations, injection-similarity and completeinjection-similarity, introduced long ago by Sz.-Nagy and Foias in [13].
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