Let ℋ denote a separable,
infinite dimensional complex Hilbert space, and let ℒ(ℋ) denote the algebra of all
bounded linear operators on ℋ. An operator X in ℒ(ℋ) is a quasiaffinity (or a
quasi-invertible operator) if X is injective and has dense range. An operator A on ℋ
is a quasiaffine transform of operator B if there exists a quasiaffinity such that
BX = XA. A and B are quasisimilar if they are quasiaffine transforms of one
another. The purpose of this note is to study the quasisimilarity orbits of
certain subsets of ℒ(ℋ) containing quasinilpotent, spectral, and compact
operators.
