An example of weakly amenable and character amenable operator

TL;DR

This paper provides a counterexample of a compact operator on an infinite-dimensional Hilbert space that is both weakly amenable and character amenable but not similar to a normal operator, addressing an open question.

Contribution

It constructs a specific example of a compact triangular operator that is weakly amenable and character amenable yet not similar to a normal operator, advancing understanding in operator algebra theory.

Findings

Counterexample of a compact operator with specific properties

Shows the operator is not similar to any normal operator

Addresses an open question in the theory of weakly amenable operators

Abstract

A complete characterization of Hilbert space operators that generate weakly amenable algebras remains open, even in the case of compact operator. Farenick, Forrest and Marcoux proposed the question that if $T$ is a compact weakly amenable operator on a Hilbert space $\mathfrak{H}$, then is $T$ similar to a normal operator? In this paper we demonstrate an example of compact triangular operator on infinite-dimension Hilbert space which is a weakly amenable and character amenable operator but is not similar to a normal operator.