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 $Lcolon\; D(L)\; ightarrow\; X$ 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 $L\_lambda$
# exists;
# is a bounded linear operator;
# is defined on a dense subspace of "X".The resolvent set of "L" is the set of all regular values of "L":

:$ho\; (L)\; =\; \{\; lambda\; in\; mathbf\{C\}\; |\; lambda\; mbox\{\; is\; a\; regular\; value\; of\; \}\; L\; \}.$

The spectrum is the complement of the resolvent set:

:$sigma\; (L)\; =\; mathbf\{C\}\; setminus\; ho\; (L).$

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 $ho(L)\; subseteq\; mathbb\{C\}$ of a bounded linear operator "L" is an open set.

References

* cite book
last = Renardy
first = Michael
coauthors = Rogers, Robert C.
title = An introduction to partial differential equations
series = Texts in Applied Mathematics 13
edition = Second edition
publisher = Springer-Verlag
address = New York
year = 2004
pages = pp. xiv+434
isbn = 0-387-00444-0 MathSciNet|id=2028503 (See section 7.3)

External links

* springer
id = R/r081610
title = Resolvent set
last = Voitsekhovskii
first = M.I.