Resolvent set

Summary

The resolvent set of an element $a$ of a unital algebra is the set of scalars $\lambda$ for which $a - \lambda 1$ is invertible. The same notion applies to linear operators. It is the complement of the spectrum. On it the resolvent $\lambda \mapsto (a - \lambda 1)^{-1}$ is defined. In a complex Banach algebra the resolvent set is open, and the resolvent is holomorphic on it. These properties of the resolvent are the main tools to study the spectrum.