  Let $\Delta$ be a linear differential operator acting on the space of
densities of a given weight $\lo$ on a manifold $M$. One can consider a pencil
of operators $\hPi(\Delta)=\{\Delta_\l\}$ passing through the operator $\Delta$
such that any $\Delta_\l$ is a linear differential operator acting on densities
of weight $\l$. This pencil can be identified with a linear differential
operator $\hD$ acting on the algebra of densities of all weights. The existence
of an invariant scalar product in the algebra of densities implies a natural
decomposition of operators, i.e. pencils of self-adjoint and anti-self-adjoint
operators. We study lifting maps that are on one hand equivariant with respect
to divergenceless vector fields, and, on the other hand, with values in
self-adjoint or anti-self-adjoint operators. In particular we analyze the
relation between these two concepts, and apply it to the study of
$\diff(M)$-equivariant liftings. Finally we briefly consider the case of
liftings equivariant with respect to the algebra of projective transformations
and describe all regular self-adjoint and anti-self-adjoint liftings.
