Resolvent set
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 [math]\displaystyle{ L\colon D(L)\rightarrow X }[/math] be a linear operator with domain [math]\displaystyle{ D(L) \subseteq X }[/math]. Let id denote the identity operator on X. For any [math]\displaystyle{ \lambda \in \mathbb{C} }[/math], let
- [math]\displaystyle{ L_{\lambda} = L - \lambda\,\mathrm{id}. }[/math]
A complex number [math]\displaystyle{ \lambda }[/math] is said to be a regular value if the following three statements are true:
- [math]\displaystyle{ L_\lambda }[/math] is injective, that is, the corestriction of [math]\displaystyle{ L_\lambda }[/math] to its image has an inverse [math]\displaystyle{ R(\lambda, L) }[/math];
- [math]\displaystyle{ R(\lambda,L) }[/math] is a bounded linear operator;
- [math]\displaystyle{ R(\lambda,L) }[/math] is defined on a dense subspace of X, that is, [math]\displaystyle{ L_\lambda }[/math] has dense range.
The resolvent set of L is the set of all regular values of L:
- [math]\displaystyle{ \rho(L) = \{ \lambda \in \mathbb{C} \mid \lambda \mbox{ is a regular value of } L \}. }[/math]
The spectrum is the complement of the resolvent set:
- [math]\displaystyle{ \sigma (L) = \mathbb{C} \setminus \rho (L). }[/math]
The spectrum can be 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).
If [math]\displaystyle{ L }[/math] is a closed operator, then so is each [math]\displaystyle{ L_\lambda }[/math], and condition 3 may be replaced by requiring that [math]\displaystyle{ L_\lambda }[/math] be surjective.
Properties
- The resolvent set [math]\displaystyle{ \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.
References
- Renardy, Michael; Rogers, Robert C. (2004). An introduction to partial differential equations. Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. p. xiv+434. ISBN 0-387-00444-0. MR2028503 (See section 8.3)
External links
- Hazewinkel, Michiel, ed. (2001), "Resolvent set", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=R/r081610
See also
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