Examples of Noncompact Quasinilpotent Hankel Operators

Abstract

For every nonzero real number $a,$ we prove that the integral Hankel operator on $L^2({\mathbb{R}}_{+})$ with kernel $k(t)=t^{-1+ia}$ is noncompact quasinilpotent. This answers in the affirmative a question by Peller \cite{Peller1998} about the existence of noncompact quasinilpotent Hankel operators. Moreover, we obtain the corresponding Hankel operators on the Hardy space with symbols ${\left((1-z)/(1+z)\right)}^{ia}.$