Operator pencil passing through a given operator

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.