Compact Operators on a Hilbert Space

Abstract

In the previous chapter we decomposed self-adjoint, normal, and unitary operators into integrals over their spectra. We should like to apply these results, especially in the self-adjoint case, to ordinary and partial differential operators. Unfortunately differential operators on the standard L2 spaces are not bounded, and thus do not satisfy the hypotheses made previously. To circumvent this difficulty in part we apply the results to the inverse operators, which are not only bounded, but in addition are of a very special form. They are compact.