We study a real hypersurface M satisfying $L_{xi}S=0;and;R_{xi}S=SR_{xi}$ in a complex hyperbolic space $H_nmathbb{C}$, where S is the Ricci tensor of type (1,1) on M, $L_{xi};and;R_{xi}$ denotes the operator of the Lie derivative and the Jacobi operator with respect to the structure vector field e respectively.