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}}.

A complex number $\lambda$ is said to be a regular value if

$L_{\lambda }$ is injective, that is, the corestriction of $L_{\lambda }$ to its image has an inverse $R(\lambda ,L)$ , which:
is a bounded linear operator;
is defined on a dense subspace of X, that is, $L_{\lambda }$ has dense range.

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).

If $L$ is a closed operator, then so is each $L_{\lambda }$ , and condition 3 may be replaced by requiring that $L_{\lambda }$ is surjective.

Properties

The resolvent set $\rho (L)\subseteq {\mathbb {C}}$ 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. xiv+434. ISBN 0-387-00444-0. MR 2028503 (See section 8.3)

External links

Voitsekhovskii, M.I. (2001) [1994], "Resolvent set", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

Resolvent set

Definitions

Properties

References

External links

See also