Linear continuous surjections of Cp-spaces over compacta

Let X and Y be compact Hausdorff spaces and suppose that there exists a linear continuous surjection T:Cp(X)→Cp(Y), where Cp(X) denotes the space of all real-valued continuous functions on X endowed with the pointwise convergence topology. We prove that dim⁡X=0 implies dim⁡Y=0. This generalizes a previous theorem [7, Theorem 3.4] for compact metrizable spaces. Also we point out that the function space Cp(P) over the pseudo-arc P admits no densely defined linear continuous operator Cp(P)→Cp([0,1]) with a dense image.