Codazzi type $h$-operators on real hypersurfaces in nonflat complex space forms

In this paper, we prove that if the operator $h$ of a real hypersurface $M$ in a nonflat complex space form $\widetilde{M}_n(c)$, $n\ge2$, is of Codazzi type, then $M$ is a Hopf hypersurface. We also prove that such a Hopf hypersurface is locally congruent to a real hypersurface of type $(A)$.