Riesz Idempotents

Let $A \in B(X)$, $X$ Banach. Let $\Delta$ be a connected component of $\sigma(A)$. Assume $\Delta \neq \sigma(A)$.
Let

$$E(\Delta) := \frac{1}{2\pi i} \int_\Gamma (z - A)^{-1} dz$$

where $\Gamma$ is a positively oriented system with $\Delta$ inside and $\sigma(A) \setminus \Delta$ outside. $E(\Delta)$ is called a Riesz Idempotent.
Let

$$X_\Delta = E(\Delta) X, \, \sigma(A\vert _{X_\Delta}) = \Delta.$$

Let

$$\Omega = \sigma(A) \setminus \Delta$$

and define $X_\Omega$ similarly. There is $R:X \rightarrow X_\Delta \oplus X_\Omega$ such that

$$RAR^{-1} = \begin{bmatrix}A\vert _{X_\Delta} & 0 \\ 0 & A\vert _{X_\Omega} \end{bmatrix}$$