Characterization of operators satisfying non-automorphic right covariance

Characterize all bounded operators A on H^2(cal E) satisfying the intertwining relation M_z A = A M_f when f is a holomorphic self-map of the unit disk that is not an automorphism.

Background

The paper proves that if a bounded operator A on H^2(\mathcal E) satisfies the right covariance relation M_zA=Af(M_z), then AC_f=M_\varphi for some \varphi\in H^\infty(\mathcal B(\mathcal E)). The authors note that this result determines the intertwining operator only on \operatorname{Ran} C_f, limiting its applicability.

If C_f has a right inverse, then A can be written as M_\varphi R_f. Since a composition operator induced by a nonconstant holomorphic self-map is injective, surjectivity—and hence invertibility—would be required for such a right inverse. The cited characterization implies that C_f is invertible exactly when f is an automorphism of the disk, leaving the general non-automorphic case unresolved.