In the first three sections of this chapter, we recall a number of standard results from functional analysis and elliptic PDE theory. The majority of these results are stated without proof; the reader is referred to the notes at the end of the chapter.

The spectral theory of compact operators

Let B be a Banach space and let A be a linear operator defined on a dense subspace D ⊂ B and taking values in B. The operator A is called closed if its graph {(x, Ax); x ∈ D) is a closed subset of B × B. It is called closable if it can be extended to a closed operator. Clearly, A is closeable if and only if, whenever xn → 0 and Axn converges, then, in fact, Axn → 0. It follows from the closed graph theorem that a closed operator defined on a Banach space B is bounded. Thus, if A is closed and unbounded, its domain of definition D must necessarily be a proper subspace of B. The resolvent set ρ(A) of a closed densely defined operator A is defined as the collection of complex numbers λ for which λ – A is a bijection of D onto B. If λ ∈ ρ(A), then (λ − A)−1 is a closed operator from B into B and it thus follows from the closed graph theorem that (λ − A)−1 is in fact a bounded operator. The resolvent set can be shown to be an open subset of ℂ.