In this paper, we investigate a real hypersurface of a complex projective space $CP^{n}$ in terms of the Jacobi operators. We give a local stmcture theorem of a real hypersurface of $CP^{n}$ satisisfying $R_{\xi}=k(I-\eta\otimes\xi)$, where $ R_{\xi}=R(\cdot, \xi)\xi$ the Jacobi operator with respect to $\xi$ and $k$ is a function. Further, we classify real hypersurfaces of $CP^{n}$ satisfying $\phi R_{\xi}=R_{\xi}\phi$ under the condition that $ A\xi$ is a principal curvature vector. Also, we show that a complex projective space does not admit a locally symmetric real hypersurface.