  Let $M$ be a real hypersurface in complex projective space. The almost
contact metric structure on $M$ allows us to consider, for any nonnull real
number $k$, the corresponding $k$-th generalized Tanaka-Webster connection on
$M$ and, associated to it, a differential operator of first order of Lie type.
Considering such a differential operator and Lie derivative we define, from the
structure Jacobi operator $R_{\xi}$ on $M$ a tensor field of type (1,2),
$R_{{\xi}_T}^{(k)}$. We obtain some classifications of real hypersurfaces for
which $R_{{\xi}_T}^{(k)}$ is either symmetric or skew symmetric.
