Notation about a linear map

Question

Let $E=\mathbb{R}^n$ and $F$ a linear subspace of $E$. Then we have $E=F \oplus F^{\perp}$.

Is it clear (and correct) if one defines : $s:=Id|_{F}+\tilde s$ ? (Where $\tilde s \in \mathcal{L}(F^\perp,F^\perp$)

To me that notation is quite ambiguous because I tend to understand it like this :

$$(Id|_{F}+\tilde s)(x)=Id|_{F}(x)+\tilde s(x).$$

That way of thinking is obviously false if the notation is correct since $Id|_{F}$ and $\tilde s$ are not defined on the same spaces. So one should read : $$(Id|_{F}+\tilde s)(x)=Id|_{F}(x_F)+\tilde s(x_{F^\perp}).$$

So :

1. Is the notation ok ?
2. If not, is there a better and simple notation ?

Reference (in french) at the beginning of the second page : here

Answer 1

Then there is no ambiguity at all: any $\;v\in\Bbb R^n\;$ is uniquely expressible as $\;v= f+\tilde f\;,\;\;f\in F\;,\;\;\tilde f\in F^\perp\;$ , so that

$$sv=\left(Id_F+\tilde s\right)(f+\tilde f)=Id_Ff+Id_F\tilde f+\tilde sf+\tilde s\tilde f=f+0+0+\tilde s\tilde f=f+\tilde s\tilde f$$

Answer 2

Your map splits as$$E\rightarrow F\oplus F^\bot\rightarrow F\oplus F^\bot\rightarrow E$$ $$x\mapsto (x_F,x_{F^\bot})\mapsto (x_F,\tilde {s}(x_{F^\bot}))\mapsto x_F+\tilde {s}(x_{F^\bot}),$$ and is indeed linear because projections are linear and composition of linear maps is linear.

In this way the map is unambiguously defined, but the notation is not very economical.

It might be helpful to provide a reference for your question.