The Spectrum of a Bounded Operator

Here we define the key objects of spectral theory: the spectrum and the resolvent, and prove that the resolvent is analytic outside the spectrum. We start by recalling a few statements about invertibility of operators.

Invertible Operators

Definition 4.1: An operator $A \in \mathcal{B}(X,Y)$ is said to be invertible if there exists an operator $A^{-1} \in \mathcal{B}(Y,X)$ (called the inverse of $A$) such that:

Proposition 4.2: _Let $A \in \mathcal{B}(X,Y)$. If there exists a left