On similarity of quasinilpotent operators

TL;DR

This paper characterizes operators similar to the Volterra operator on Banach spaces, showing that such similarity does not necessarily determine the underlying space's topology, with implications for operators on Hardy spaces.

Contribution

It provides a characterization of operators similar to the Volterra operator and establishes conditions under which an operator determines the topology of a Banach space.

Findings

Operators similar to the Volterra operator do not determine the topology of $C[0,1]$.

A sufficient condition is given for an operator to determine the space's topology.

The Volterra operator on Hardy spaces $ ext{H}^p$ does determine the topology for all $p \

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 $p\in[1,\infty]$.