  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.
