Resolvent

A resolvent is a concept in functional analysis associated with a linear operator on a Banach space. If A is a linear operator with domain D(A) in a complex Banach space X, the resolvent set ρ(A) consists of all complex numbers λ for which A − λI is bijective from D(A) onto X and has a bounded inverse. That inverse, R(λ, A) = (A − λI)^{-1}, is called the resolvent operator. The complement of ρ(A) in the complex plane is the spectrum σ(A). The spectrum is the set of λ for which A − λI fails to be invertible or its inverse is unbounded.

The resolvent set is open, and the resolvent operator is analytic as a function of λ on ρ(A). A fundamental identity, the resolvent identity, states that for any λ, µ in ρ(A),

R(λ, A) − R(µ, A) = (λ − µ) R(λ, A) R(µ, A).

This yields many analytical consequences, including a relation between the spectrum and the growth of the resolvent.

Several basic properties accompany these definitions. For a bounded operator A, the resolvent set contains all λ with sufficiently large modulus outside the spectrum, and for such λ one has the bound

||R(λ, A)|| ≤ 1 / dist(λ, σ(A)).

Near infinity, the resolvent admits the expansion R(λ, A) ~ −(1/λ) I − (1/λ^2) A − ... (the Neumann series gives an explicit expansion when |λ| > ||A||).

Special cases and applications. In finite dimensions, the resolvent is (A − λI)^{-1} for every λ not an eigenvalue, with poles at eigenvalues. In analysis and PDEs, the resolvent is central to spectral theory, semigroup theory, and the solution of linear equations via Laplace transforms, linking resolvent operators to time-evolution and Green’s functions. The spectrum further partitions into point, continuous, and residual components, informing the operator’s behavior.