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In [[linear algebra]] and [[operator theory]], the '''resolvent set''' of a [[linear operator]] is a [[Set (mathematics)|set]] of [[complex number]]s for which the operator is in some sense "[[Pathological_(mathematics)#Well-behaved|well-behaved]]".  The resolvent set plays an important role in the [[resolvent formalism]].

==Definitions==

Let ''X'' be a [[Banach space]] and let <math>L\colon D(L)\rightarrow X</math> be a linear operator with [[Domain of a function|domain]] <math>D(L) \subseteq X</math>.  Let id denote the [[identity function|identity operator]] on ''X''.  For any <math>\lambda \in \mathbb{C}</math>, let

:<math>L_{\lambda} = L - \lambda\,\mathrm{id}.</math>

A complex number <math>\lambda</math> is said to be a '''regular value''' if the following three statements are true:
# <math>L_\lambda</math> is [[injective]], that is, the corestriction of <math>L_\lambda</math> to its image has an [[Inverse function|inverse]] <math>R(\lambda, L)=(L-\lambda \,\mathrm{id})^{-1}</math> called the '''resolvent''';{{sfn | Reed | Simon | 1980 | p=188}}
# <math>R(\lambda,L)</math> is a [[bounded linear operator]];
# <math>R(\lambda,L)</math> is defined on a [[Dense set|dense]] [[Linear subspace|subspace]] of ''X'', that is, <math>L_\lambda</math> has dense range.
The '''resolvent set''' of ''L'' is the set of all regular values of ''L'':

:<math>\rho(L) = \{ \lambda \in \mathbb{C} \mid \lambda \mbox{ is a regular value of } L \}.</math>

The [[Spectrum (functional analysis)|spectrum]] is the [[Complement (set theory)|complement]] of the resolvent set

:<math>\sigma (L) = \mathbb{C} \setminus \rho (L),</math>

and subject to a mutually singular [[Decomposition_of_spectrum_(functional_analysis)|spectral decomposition]] into the [[Spectrum_(functional_analysis)#Point_spectrum|point spectrum]] (when condition 1 fails), the [[Spectrum_(functional_analysis)#Continuous_spectrum|continuous spectrum]] (when condition 2 fails) and the [[Spectrum_(functional_analysis)#Residual_spectrum|residual spectrum]] (when condition 3 fails).

If <math>L</math> is a [[closed operator]], then so is each <math>L_\lambda</math>, and condition 3 may be replaced by requiring that <math>L_\lambda</math> be [[surjective]].

==Properties==

* The resolvent set <math>\rho(L) \subseteq \mathbb{C}</math> 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==
{{reflist}}

==References==
* {{cite book | last1=Reed | first1=M. | last2=Simon | first2=B. | title=Methods of Modern Mathematical Physics: Vol 1: Functional analysis | publisher=Academic Press | year=1980 | isbn=978-0-12-585050-6}}
* {{cite book
| last = Renardy
| first = Michael
|author2=Rogers, Robert C.
 | title = An introduction to partial differential equations
| series = Texts in Applied Mathematics 13
| edition = Second
| publisher = Springer-Verlag
| location = New York
| year = 2004
| isbn = 0-387-00444-0
| page = xiv+434
| no-pp = true
}} {{MathSciNet|id=2028503}} (See section 8.3)

==External links==

* {{springer|
| id = R/r081610
| title = Resolvent set
| last = Voitsekhovskii
| first = M.I.
}}

==See also==
*[[Resolvent formalism]]
*[[Spectrum (functional analysis)]]
*[[Decomposition of spectrum (functional analysis)]]


[[Category:Linear algebra]]
[[Category:Operator theory]]