Examples of tauberian operators acting on C[0,1]

We show that some counterexamples in the theory of tauberian operators can be realized as operators acting on C[0,1]. Precisely, we show that the set τ(C[0,1]) of tauberian operators acting on C[0,1] is not open, and that T∈τ(C[0,1]) does not imply T⁎⁎ tauberian.