On Harrell-Stubbe Type Inequalities for the Discrete Spectrum of a Self-Adjoint Operator

We produce a new proof and extend results by Harrell and Stubbe for the discrete spectrum of a self-adjoint operator. An abstract approach–based on commutator algebra, the Rayleigh-Ritz principle, and an optimal'' usage of the Cauchy-Schwarz inequality--is used to produce parameter-free’‘, ``projection-free’’ versions of their theorems. We also analyze the strength of the various inequalities that ensue. The results contain classical bounds for the eigenvalues. Extensions of a variety of inequalities `a la Harrell-Stubbe are illustrated for both geometric and physical problems.