On a problem of H.P. Rosenthal concerning operators on C[0,1]

A problem of H.P. Rosenthal asks whether every bounded linear operator which is an isomorphism on a closed linear infinite-dimensional subspace X not containing any isomorph of c0, is actually an isomorphism on a subspace isomorphic to C[0,1]. An affirmative answer to this problem is provided when T is a contraction whose restriction to X is an isometry.