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 \lambda is said to be a regular value if the following three statements are true: The resolvent set of L is the set of all regular values of L: The spectrum is the complement of the resolvent set 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 L is a closed operator, then so is each L_\lambda, and condition 3 may be replaced by requiring that L_\lambda be surjective.

Properties