  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.
