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 be a linear operator with domain. Let id denote the identity operator on X. For any, let
A complex number is said to be a regular value if
  1. is injective, that is, the corestriction of to its image has an inverse, which:
  2. is a bounded linear operator;
  3. is defined on a dense subspace of X, that is, has dense range.
The resolvent set of L is the set of all regular values of L:
The spectrum is the complement of the resolvent set:
The spectrum can be further decomposed into the point/discrete spectrum, the continuous spectrum and the residual/compression spectrum.
If is a closed operator, then so is each, and condition 3 may be replaced by requiring that is surjective.

Properties