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In [[linear algebra]] and [[operator theory]], the '''resolvent set''' of a [[linear operator]] is a set of [[complex number]]s 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 <math>L\colon D(L)\rightarrow X</math> be a linear operator with [[domain (mathematics)|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> | |||
<math>\lambda</math> is said to be a '''regular value''' if <math>R(\lambda, L)</math>, the [[inverse function|inverse operator]] to <math>L_\lambda</math> | |||
# exists; | |||
# is a [[bounded linear operator]]; | |||
# is defined on a [[dense set|dense]] subspace of ''X''. | |||
The '''resolvent set''' of ''L'' is the set of all regular values of ''L'': | |||
:<math>\rho (L) = \{ \lambda \in \mathbb{C} | \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> | |||
The spectrum can be further decomposed into the point/discrete spectrum (where condition 1 fails), the continuous spectrum (where conditions 1 and 3 hold but condition 2 fails) and the residual/compression spectrum (where condition 1 holds but condition 3 fails). | |||
==Properties== | |||
* The resolvent set <math>\rho(L) \subseteq \mathbb{C}</math> of a bounded linear operator ''L'' is an [[open set]]. | |||
==References== | |||
* {{cite book | |||
| last = Renardy | |||
| first = Michael | |||
| coauthors = Rogers, Robert C. | |||
| title = An introduction to partial differential equations | |||
| series = Texts in Applied Mathematics 13 | |||
| edition = Second | |||
| publisher = Springer-Verlag | |||
| address = New York | |||
| year = 2004 | |||
| isbn = 0-387-00444-0 | |||
| page = xiv+434 | |||
| nopp = true | |||
}} {{MathSciNet|id=2028503}} (See section 8.3) | |||
==External links== | |||
* {{springer| | |||
| id = R/r081610 | |||
| title = Resolvent set | |||
| last = Voitsekhovskii | |||
| first = M.I. | |||
}} | |||
[[Category:Linear algebra]] | |||
[[Category:Operator theory]] | |||
{{Linear-algebra-stub}} | |||
Latest revision as of 18:29, 28 September 2013
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 be a linear operator with domain . Let id denote the identity operator on X. For any , let
is said to be a regular value if , the inverse operator to
- exists;
- is a bounded linear operator;
- is defined on a dense subspace of X.
The resolvent set of L is the set of all regular values of L:
The spectrum is the complement of the resolvent set:
The spectrum can be further decomposed into the point/discrete spectrum (where condition 1 fails), the continuous spectrum (where conditions 1 and 3 hold but condition 2 fails) and the residual/compression spectrum (where condition 1 holds but condition 3 fails).
Properties
- The resolvent set of a bounded linear operator L is an open set.
References
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External links
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