An example of a nonzero quasinilpotent operator with reflexive commutant is presented.

A new example of a non-zero quasi-nilpotent operator T with reflexive commutant is presented. The norms $|T^n|$ converge to zero arbitrarily fast.

Let T be a spherical 2-expansive m-tuple and let $T^{}$ denote its spherical Cauchy dual. If $T^{}$ is commuting then the inequality $\genfrac{}{}{0pt}{}{{\sum }_{\left|\beta \right|=k}{(\beta !)}^{-1}{\left(T^{}\right)}^{\beta }\left(T^{}\right){*}^{\beta }\le (k+m-1}{k){\sum }_{\left|\beta \right|=k}{(\beta !)}^{-1}\left(T^{}\right){*}^{\beta }{\left(T^{}\right)}^{\beta }}$ holds for every positive integer k. In case m = 1, this reveals the rather curious fact that all positive integral powers of the Cauchy dual of a 2-expansive (or concave) operator are hyponormal.