User:Caliburn/s/fa/Definition:Resolvent of Linear Operator

Definition
Let $\struct {X, \norm \cdot_X}$ be a Banach space over $\C$.

Let $A : X \to X$ be a linear operator.

Let $\map \rho A$ be the resolvent set of $A$.

Let $\lambda \in \map \rho A$.

Let:


 * $R_\lambda = A - \lambda I$

We call $R_\lambda$ a resolvent of $A$.