Examples of tauberian operators acting on

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.