Almost every quasinilpotent Hilbert space operator is a universal quasinilpotent

Abstract:

Let Q be a quasinilpotent operator acting on a complex separable infinite dimensional Hilbert space; then either ${Q^k}$ is compact for some positive integer k, or the closure of the similarity orbit of Q contains every quasinilpotent operator. Analogous results are shown to be true for the Calkin algebra and for nonseparable Hilbert spaces. For the nonseparable case, the analogous result is true for the closed bilateral ideal $\mathcal {J}$, strictly larger than the ideal of compact operators, if and only if $\mathcal {J}$ is not the ideal associated with an ${\aleph _0}$-regular limit cardinal. For the ideal of compact operators, the problem remains open.