Resolvents

Resolvents are a fundamental concept in functional analysis and operator theory. For a linear operator T on a Banach space, a complex number λ lies in the resolvent set ρ(T) if T − λI is bijective and its inverse (T − λI)−1 is a bounded operator. The spectrum σ(T) is the complement of ρ(T) in the complex plane. The definition extends to unbounded operators in the same spirit, with the inverse required to exist on the whole space and be bounded.

The resolvent operator is defined by R(λ; T) = (T − λI)−1 for λ in ρ(T). It depends analytically on λ; the map λ ↦ R(λ; T) is holomorphic on ρ(T) with values in the bounded operators. A key relation is the resolvent identity: R(λ; T) − R(μ; T) = (μ − λ) R(λ; T) R(μ; T) for λ, μ in ρ(T). As λ approaches the spectrum, the norm of the resolvent tends to infinity, reflecting the failure of invertibility.

In finite dimensions, T is represented by a matrix and σ(T) consists of its eigenvalues; ρ(T) is the complement in the complex plane, and R(λ; T) = (T − λI)−1 when defined. For self-adjoint or normal operators on Hilbert spaces, the spectrum lies on the real axis (or the spectrum is contained in the real axis for self-adjoint operators), and the resolvent is well-behaved off this set.

Applications of resolvents include the spectral theorem and functional calculus, analysis of evolution equations via semigroups, and perturbation theory. They provide tools for contour integral representations of functions of operators and for understanding how small changes in an operator affect its spectrum.