On similarity of quasinilpotent operators

Abstract

Bounded linear operators on separable Banach spaces algebraically similar to the classical Volterra operator $V$ acting on $C[0,1]$ are characterized. From this characterization it follows that $V$ does not determine the topology of $C[0,1]$, which answers a question raised by Armando Villena. A sufficient condition for an injective bounded linear operator on a Banach space to determine its topology is obtained. From this condition it follows, for instance, that the Volterra operator acting on the Hardy space $\H^p$ of the unit disk determines the topology of $\H^p$ for any $p\in[1,\infty]$.