I'm reading Difference Operators and Summation Formulas from textbook Analysis I by Amann/Escher.
The authors omit some detail and I would like to verify whether my understanding is correct or not.
Let $E$ be a vector space over field $\mathbb{K}$. To simplify notation, we write ${\operatorname{G}f}_n$ for $(\operatorname{G}(f))_n$ where $\operatorname{G} \in \operatorname{End}\left(E^{\mathbb{N}}\right)$ and $f \in E^{\mathbb{N}}$.
On $E^{\mathbb{N}}$ define operators $\triangle, \operatorname{I}, \operatorname{O}$ by $$\triangle f_{n} :=f_{n+1}-f_{n} \text{ and } \operatorname{I} f_{n} := f_{n} \text{ and } \operatorname{O} f_{n} := 0$$
It is easy to verify that $\triangle, \operatorname{I} ,\operatorname{O} \in \operatorname{End}\left(E^{\mathbb{N}}\right)$.
On $\operatorname{End}\left(E^{\mathbb{N}}\right)$ define addition $+$ and multiplication $\cdot$ by $$(\operatorname{G} + \operatorname{H})f_n := {\operatorname{G}f}_n + {\operatorname{H}f}_n \text{ and } (\operatorname{G} \cdot \operatorname{H})f := \operatorname{G} (\operatorname{H}f), \quad G,H \in \operatorname{End}\left(E^{\mathbb{N}}\right)$$
As such, $\left \langle \operatorname{End}\left(E^{\mathbb{N}}\right), +, \cdot \right \rangle$ is a (not necessarily commutative) ring with unity. More specifically, $\operatorname{I}$ is the multiplicative identity and $\operatorname{O}$ the additive identity of this ring. As such, $\triangle \cdot \operatorname{I} = \operatorname{I} \cdot \triangle$.
Thank you for your help!
You understood correctly ! As stated in the comments (if you know what that means) $(\lambda\cdot G) f_n := \lambda Gf_n$ also defines a $\mathbb K$-algebra structure on $\mathrm{End}(E^\mathbb N)$.
In fact this works for any vector space $V$, not just for $E^\mathbb N$