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 ${\displaystyle L\colon D(L)\rightarrow X}$  be a linear operator with domain ${\displaystyle D(L)\subseteq X}$ . Let id denote the identity operator on X. For any ${\displaystyle \lambda \in \mathbb {C} }$ , let

${\displaystyle L_{\lambda }=L-\lambda \,\mathrm {id} .}$ 

A complex number ${\displaystyle \lambda }$  is said to be a regular value if the following three statements are true:

1. ${\displaystyle L_{\lambda }}$  is injective, that is, the corestriction of ${\displaystyle L_{\lambda }}$  to its image has an inverse ${\displaystyle R(\lambda ,L)=(L-\lambda \,\mathrm {id} )^{-1}}$  called the resolvent;^[1]
2. ${\displaystyle R(\lambda ,L)}$  is a bounded linear operator;
3. ${\displaystyle R(\lambda ,L)}$  is defined on a dense subspace of X, that is, ${\displaystyle L_{\lambda }}$  has dense range.

The resolvent set of L is the set of all regular values of L:

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

The spectrum is the complement of the resolvent set

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

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

Properties

• The resolvent set ${\displaystyle \rho (L)\subseteq \mathbb {C} }$  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

1. Reed & Simon 1980, p. 188.

References

• Reed, M.; Simon, B. (1980). Methods of Modern Mathematical Physics: Vol 1: Functional analysis. Academic Press. ISBN 978-0-12-585050-6.
• 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. MR2028503 (See section 8.3)

External links

• Voitsekhovskii, M.I. (2001) [1994], "Resolvent set", Encyclopedia of Mathematics, EMS Press

See also

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