Universal bounds and semiclassical estimates for eigenvalues of abstract Schroedinger operators

We prove trace inequalities for a self-adjoint operator on an abstract Hilbert space. These inequalities lead to universal bounds on spectral gaps and on moments of eigenvalues lambda_k that are analogous to those known for Schroedinger operators and the Dirichlet Laplacian, on which the operators of interest are modeled. In addition we produce inequalities that are new even in the model case. These include a family of differential inequalities for generalized Riesz means and theorems stating that arithmetic means of lambda_k^p for p <= 3 are universally bounded from above by multiples of the geometric mean of the lambda_k. For Schroedinger operators and the Dirichlet Laplacian these bounds are Weyl-sharp, i.e., saturated by the standard semiclassical estimates for lambda_k at large k.