  In this paper, we obtain a new abstract formula relating eigenvalues of a
self-adjoint operator to two families of symmetric and skew-symmetric operators
and their commutators. This formula generalizes earlier ones obtained by
Harrell, Stubbe, Hook, Ashbaugh, Hermi, Levitin and Parnovski. We also show how
one can use this abstract formulation both for giving dierent and simpler
proofs for all the known results obtained for the eigenvalues of a power of the
Laplace operator (i.e. the Dirichlet Laplacian, the clamped plate problem for
the bilaplacian and more generally for the polyharmonic problem on a bounded
Euclidean domain) and to obtain new ones. In a last paragraph, we derive new
bounds for eigenvalues of any power of the Kohn Laplacian on the Heisenberg
group.
