I give a geometric characterization of mean ergodic convergence in the Calkin algebras for Banach spaces that have the bounded compact approximation property; I 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) affirmative answers to some questions of Hadwin, and (iii) an alternative proof of Hadwin's characterization of the strong, weak and *-strong operator topologies of the unitary orbit of a given operator on a separable, infinite dimensional, complex Hilbert space; I study appropriately normalized square random Vandermonde matrices based on independent random variables with uniform distribution on the unit circle; and I show that as the matrix size increases without bound, with respect to the expectation of the trace there is an asymptotic *-distribution, equal to that of a C[0, 1]-valued R-diagonal element.