---
tag: X0243CZD9D
kind: concept
name: Resolvent set
msc: [47A10, 46H05]
sources:
  - url: https://en.wikipedia.org/wiki/Resolvent_set
same_as:
  wikidata: Q7315720
  wikipedia: 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.

## Explanations

### L1 human

The resolvent set collects the numbers $\lambda$ for which the equation $(a - \lambda) x = y$ can always be solved in a unique and stable way.
For a matrix, these are all numbers that are not eigenvalues.
For multiplication by the function $t$ on $[0,1]$, these are the complex numbers outside the interval $[0,1]$.