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 $L\colon D(L)\rightarrow X$ be a linear operator with domain $D(L)\subseteq X$. Let id denote the identity operator on X. For any $\lambda \in \mathbb {C}$, let

$L_{\lambda }=L-\lambda \mathrm {id} .$

$\lambda$ is said to be a regular value if $R(\lambda ,L)$, the inverse operator to $L_{\lambda }$

1. exists, that is, $L_{\lambda }$ is injective;
2. is a bounded linear operator;
3. is defined on a dense subspace of X.

The resolvent set of L is the set of all regular values of L:

$\rho (L)=\{\lambda \in \mathbb {C} |\lambda {\mbox{ is a regular value of }}L\}.$

The spectrum is the complement of the resolvent set:

$\sigma (L)=\mathbb {C} \setminus \rho (L).$

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 $\rho (L)\subseteq \mathbb {C}$ of a bounded linear operator L 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. xiv+434. ISBN 0-387-00444-0. {{cite book}}: Unknown parameter |nopp= ignored (|no-pp= suggested) (help) MR2028503 (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)