In linear algebra and operator theory, the resolvent set of a linear operator is a set of complex numbers for which the operator is in some sense "well-behaved". The resolvent set plays an important role in the resolvent formalism.
Definitions Let X be a Banach space and let L : D ( L ) → X {\displaystyle L\colon D(L)\rightarrow X} be a linear operator with domain D ( L ) ⊆ X {\displaystyle D(L)\subseteq X} . Let id denote the identity operator on X. For any λ ∈ C {\displaystyle \lambda \in \mathbb {C} } , let
L λ = L − λ i d . {\displaystyle L_{\lambda }=L-\lambda \,\mathrm {id} .}
A complex number λ {\displaystyle \lambda } is said to be a regular value if the following three statements are true:
L λ {\displaystyle L_{\lambda }} is injective, that is, the corestriction of L λ {\displaystyle L_{\lambda }} to its image has an inverse R ( λ , L ) = ( L − λ i d ) − 1 {\displaystyle R(\lambda ,L)=(L-\lambda \,\mathrm {id} )^{-1}} called the resolvent;
R ( λ , L ) {\displaystyle R(\lambda ,L)} is a bounded linear operator;
R ( λ , L ) {\displaystyle R(\lambda ,L)} is defined on a dense subspace of X, that is, L λ {\displaystyle L_{\lambda }} has dense range. The resolvent set of L is the set of all regular values of L:
ρ ( L ) = { λ ∈ C ∣ λ is a regular value of L } . {\displaystyle \rho (L)=\{\lambda \in \mathbb {C} \mid \lambda {\mbox{ is a regular value of }}L\}.}
The spectrum is the complement of the resolvent set
σ ( L ) = C ∖ ρ ( L ) , {\displaystyle \sigma (L)=\mathbb {C} \setminus \rho (L),}
and subject to a mutually singular spectral decomposition into the point spectrum (when condition 1 fails), the continuous spectrum (when condition 2 fails) and the residual spectrum (when condition 3 fails). If L {\displaystyle L} is a closed operator, then so is each L λ {\displaystyle L_{\lambda }} , and condition 3 may be replaced by requiring that L λ {\displaystyle L_{\lambda }} be surjective.
Properties The resolvent set ρ ( L ) ⊆ C {\displaystyle \rho (L)\subseteq \mathbb {C} } of a bounded linear operator L is an open set. More generally, the resolvent set of a densely defined closed unbounded operator is an open set.
Notes
References Reed, M.; Simon, B. (1980). Methods of Modern Mathematical Physics: Vol 1: Functional analysis. Academic Press. ISBN 978-0-12-585050-6. Renardy, Michael; Rogers, Robert C. (2004). An introduction to partial differential equations. Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. xiv+434. ISBN 0-387-00444-0. MR 2028503 (See section 8.3)
External links Voitsekhovskii, M.I. (2001) [1994], "Resolvent set", Encyclopedia of Mathematics, EMS Press
See also Resolvent formalism Spectrum (functional analysis) Decomposition of spectrum (functional analysis)
