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