Math 2.0

Operator theory Definition

Resolvent set

Builds on Spectrum of an elementUnit of a ringBanach space

Definition

Let 𝕂\mathbb{K} be a field, let AA be an associative algebra over 𝕂\mathbb{K} with unit 11, and let a∈Aa \in A. The resolvent set of aa is ρ(a)={λ∈𝕂:a−λ1 is invertible in A}=𝕂∖σ(a),\rho(a) = \set{\lambda \in \mathbb{K} : a - \lambda 1 \text{ is invertible in } A} = \mathbb{K} \setminus \sigma(a), where invertible means being a unit of AA, and σ(a)\sigma(a) is the spectrum of aa. For λ∈ρ(a)\lambda \in \rho(a), the element R(λ)=(a−λ1)−1R(\lambda) = (a - \lambda 1)^{-1} is the resolvent of aa at λ\lambda.

Let TT be a closed linear operator with domain D(T)D(T) in a complex Banach space XX. The resolvent set of TT is the set of λ∈ℂ\lambda \in \C such that T−λT - \lambda maps D(T)D(T) bijectively onto XX with bounded inverse.

Remarks

Some authors use (λ1−a)−1(\lambda 1 - a)^{-1} as the resolvent; this changes its sign only. The resolvent satisfies the resolvent identity R(λ)−R(μ)=(λ−μ)R(λ)R(μ)R(\lambda) - R(\mu) = (\lambda - \mu) R(\lambda) R(\mu) for λ,μ∈ρ(a)\lambda, \mu \in \rho(a). It follows from R(λ)−R(μ)=R(λ)((a−μ1)−(a−λ1))R(μ)R(\lambda) - R(\mu) = R(\lambda) \big( (a - \mu 1) - (a - \lambda 1) \big) R(\mu).

Example: in C([0,1])C([0,1]) with complex values, let f(t)=tf(t) = t. Then ρ(f)=ℂ∖[0,1]\rho(f) = \C \setminus [0,1], and R(λ)(t)=1/(t−λ)R(\lambda)(t) = 1/(t - \lambda).

Non-example: for the same ff, the number λ=1/2\lambda = 1/2 is not in ρ(f)\rho(f), because f−1/2f - 1/2 vanishes at t=1/2t = 1/2 and has no continuous inverse.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

Show the path as a graph
Banach space, concept, no evidence levelBanach spaceconceptSpectrum of an element, concept, no evidence levelSpectrum of an elementconceptUnit of a ring, concept, no evidence levelUnit of a ringconceptResolvent set, definition, no evidence levelResolvent setdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Banach space concept
  2. Spectrum of an element concept
  3. Unit of a ring concept
  4. Resolvent set definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Resolvent_set
About this page
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Kinddefinition
WritingW0 stub
MSC47A10, 46H05
DefinesResolvent set
UsesSpectrum of an element, Unit of a ring, Banach space
Changed2 November 2026

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