Resolvent

Summary

The resolvent of an operator $T$ is the operator-valued function $\lambda \mapsto (T - \lambda I)^{-1}$, defined on the resolvent set, the complement of the spectrum. It is an analytic function of $\lambda$, and its singularities are exactly the points of the spectrum. Through the resolvent, complex analysis becomes a tool of operator theory: contour integrals of the resolvent give projections and functions of $T$.