The spectral theory of elliptic operators on smooth bounded domains
Ross G. Pinsky · Cambridge University Press eBooks · 1995
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 x n → 0 and Ax n converges, then, in fact, Ax n → 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.