Definition:Resolvent Set/Bounded Linear Operator

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

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

Let $I : X \to X$ be the identity mapping on $X$.

Let $\map \rho A$ be the set of $\lambda \in \C$ such that $A - \lambda I$ is invertible in the sense of a bounded linear transformation

We call $\map \rho A$ the resolvent set of $A$.

Also see

  • Results about resolvent sets of bounded linear operators can be found here.