Let M be a real hypersurface of a complex space form with almost contact metric structure (${phi}$, ${xi}$, ${eta}$, g). In this paper, we study real hypersurfaces in a complex space form whose structure Jacobi operator $R_{xi}=R({cdot},;{xi}){xi}$ is ${xi}$-parallel. In particular, we prove that the condition ${ abla}_{xi}R_{xi}=0$ characterizes the homogeneous real hypersurfaces of type A in a complex projective space or a complex hyperbolic space when $R_{xi}{phi}S=R_{xi}S{phi}$ holds on M, where S denotes the Ricci tensor of type (1,1) on M.