We obtain (i) a new, coordinate free, characterization of quasidiagonal
operators with essential spectra contained in the unit circle by adapting the
proof of a classical result in the theory of Banach spaces, (ii) an affirmative
answer to some questions of Hadwin, and (iii) an alternative proof of Hadwin's
characterization of the SOT, WOT and $*$-SOT closure of the unitary orbit of a
given operator on a separable, infinite dimensional, complex Hilbert space