Understanding $E(\sigma)X$

Question

I am studying a chapter on Dunford calculus and have trouble understanding the notation $E(\sigma)X$.

Let $X$ be a Banach space over $\mathbb{C}$ and $T\in\mathcal{L}(X)$; $\sigma$ denotes a spectral set (open and closed subset of $\sigma(T)$) and $E(\sigma)=E(\sigma,T)$ is defined as the function which is identically 1 on $\sigma$ and equal to 0 on $\sigma(T)\setminus\sigma$.

Does anybody know what is meant by $E(\sigma)X$?

Answer 1

$E(\sigma)X$ is the image of $X$ under the projection $ E(\sigma)$, hence

$E(\sigma)X=\{E(\sigma)x: x \in X\}$.