Operators on Hilbert Spaces

Abstract

We now continue to develop additional properties of linear operators on Hilbert spaces. The inner product structure of a Hilbert space has many consequences for the structure of operators mapping the space into itself. The most important of these is the existence of an adjoint operator acting on the same space. Although it is possible to define an adjoint operator corresponding to an operator on a Banach space, the operator acts on a different space in general. Because of the Riesz representation theorem, the adjoint of a Hilbert space operator can be taken to act on the same space. We conclude the chapter with a discussion of the resolvent of the Laplacian on ℝ³. This important partial differential operator is another example of a Schrödinger operator and will play a central role throughout the remainder of the book.