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 $\lambda \in \mathbb$, let

:$L_ = L - \lambda\,\mathrm.$

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 $R(\lambda, L)$;
# $R(\lambda,L)$ 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'':

:$\rho(L) = \.$

The spectrum is the complement of the resolvent set:

:$\sigma (L) = \mathbb \setminus \rho (L).$

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 $\rho(L) \subseteq \mathbb$ 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

* (See section 8.3)

External links

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

See also

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

Linear algebra
Operator theory