If we view (linear) operators as nothing than multipliers multiply their eigenvectors with numbers
$$Tu=λu$$
then aren't they the same with projections from a finite point in projective geometry?
There are projection operators in functional analysis projecting the whole space into its subspace, but they're not linear, while this projection is analogous to the definition of linear operator and it stretches the whole space. The Möbius transform is linear transform on homogeneous coordinates in $\Bbb C^2$.
Can I see linear operators as protections in projective geometry without trouble?
*What is the significance of differential operators over other operators in group theory?
This was too long for a comment.
Let's consider a simple example. A generic $2 \times 2$ matrix $A$ has $2$ distinct eigenvalues, so let's say $Av=\lambda v$ and $Aw=\sigma w$ with $\lambda \ne \sigma$, hence $v \ne w$. That means that $A$ dilates the eigenlines, that is, $A$ scales the span of $v$ by $\lambda$ and it scales the span of $w$ by $\sigma$. In projective space $\mathbb{P}^1$, these lines are now two fixed points. If $|\lambda|>|\sigma|$ then the span of $v$ is an attracting fixed point, while if $|\lambda|<|\sigma|$ the span of $v$ is a repelling fixed point.
In general the dynamics are a bit more subtle, because the matrix may not be diagonalizable and because the eigendirections sort of stratify. The largest eigenvalue corresponds to an attracting point (or region), while the next largest becomes a saddle point (or region).