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 ${\displaystyle \lambda \in \mathbb{C}}$, 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. MR 2028503 (See section 8.3)

External links

• Voitsekhovskii, M.I. (2001), "Resolvent set", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

See also

• Resolvent formalism
• Spectrum (functional analysis)
• Decomposition of spectrum (functional analysis)