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We consider real hypersurfaces $M$ in complex projective space equipped withboth the Levi-Civita and generalized Tanaka-Webster connections. For anynonnull constant $k$ and any symmetric tensor field of type (1,1) $L$ on $M$ wecan define two tensor fields of type (1,2) on $M$, $L_F^{(k)}$ and $L_T^{(k)}$,related to both connections. We study the behaviour of the structure operator$\phi$ with respect to such tensor fields in the particular case of $L=A$, theshape operator of $M$, and obtain some new characterizations of ruled realhypersurfaces in complex projective space.

New characterizations of ruled real hypersurfaces in complex projective space