  We introduce a new family of invariant differential operators associated with
$\lambda$-cosine and Funk-Radon transforms on Stiefel and Grassmann manifolds.
These operators reduce the order of the $\lambda$-cosine transforms and yield
new inversion formulas. Intermediate Funk-cosine transforms corresponding to
integration over matrices of lower rank are studied. The main tools are polar
decomposition and Fourier analysis on matrix space.
