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

Universal Bounds and Semiclassical Estimates for Eigenvalues of Abstract Schrödinger Operators