  We investigate the action of Alesker's Lefschetz operators on translation
invariant valuations on convex bodies. For scalar valued valuations, we
describe this action on the level of Klain-Schneider functions by a Radon type
transform, generalizing a result by Schuster and Wannerer. In the case of
rotationally equivariant Minkowski valuations, the Lefschetz operators act on
the generating function as a convolution transform. We show that the
convolution kernel satisfies a Legendre type differential equation, and thus,
is a strictly positive function that is smooth up to one point.
