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 the following three statements are true:

$L_\lambda$ is injective, that is, the corestriction of $L_\lambda$ to its image has an inverse $R(\lambda, L)=(L-\lambda \,\mathrm{id})^{-1}$ called the resolvent;{{sfn | Reed | Simon | 1980 | p=188}}
$R(\lambda,L)$ is a bounded linear operator;
$R(\lambda,L)$ 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:

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

The spectrum is the complement of the resolvent set

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

and subject to a mutually singular spectral decomposition into the point spectrum (when condition 1 fails), the continuous spectrum (when condition 2 fails) and the residual spectrum (when condition 3 fails).

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

Notes

References

{{cite book | last1=Reed | first1=M. | last2=Simon | first2=B. | title=Methods of Modern Mathematical Physics: Vol 1: Functional analysis | publisher=Academic Press | year=1980 | isbn=978-0-12-585050-6}}
{{cite book

| last = Renardy

| first = Michael

|author2=Rogers, Robert C.

| title = An introduction to partial differential equations

| series = Texts in Applied Mathematics 13

| edition = Second

| publisher = Springer-Verlag

| location = New York

| year = 2004

| isbn = 0-387-00444-0

| page = xiv+434

| no-pp = true

}} {{MathSciNet|id=2028503}} (See section 8.3)

External links

{{springer|

| id = R/r081610

| title = Resolvent set

| last = Voitsekhovskii

| first = M.I.

}}