Is the span of the eigenvectors of a non self-adjoint compact operator dense in the operator range?

I know that for a compact self-adjoint operator (I assume a separable Hilbert space), the eigenfunctions form a Schauder basis for the entire space. But if the operator is not self-adjoint, does this property still hold at least for the range of the operator? I cannot find a reference of it, which looks suspicious to me.