Resolvent set

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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[edit]

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;
  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[edit]

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

References[edit]

  • 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.  MR 2028503 (See section 8.3)

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