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

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

$\lambda$ is said to be a regular value if $R(\lambda, L)$, the inverse operator to $L_\lambda$

1. exists, that is, $L_\lambda$ is injective;
2. is a bounded linear operator;
3. is defined on a dense subspace of X.

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).

Properties

• The resolvent set $\rho(L) \subseteq \mathbb{C}$ of a bounded linear operator L is an open set.

References

• MR 2028503 (See section 8.3)

See also

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