I am thinking about the invariant subspace problem or some related problems like the almost invariant half-space problem. In this type of problems one has the following
If a statement holds for an operator $T$, then it holds for $\alpha T$, where $\alpha$ is any nonzero scalar.
For instance, if Y is invariant/ almost invariant under T then it is also invariant/ almost invariant under $\alpha T$ ($\alpha\neq 0$).
So in dealing with these problems, the only relevant properties of operators are the ones that are invariant under nonzero scalar multiplication.
For instance to find an invariant subspace of an algebra of operators, say, $\mathcal{A}\subset \mathcal{L}(X)$, one might define the equivalent relation on $\mathcal{A}$ by $T\sim S\Leftrightarrow T=\alpha S$ for some $\alpha\neq 0$. Or one might define a set $ \mathcal{A}'=\left\{\frac{T}{\|T\|}:T\in\mathcal{A}, \ell T e\ge 0\right\}, $ where $\ell$ is a fixed functional and $e$ a fixed elements in $X$.
We might even define addition and multiplication on this set in the natural way though addition fails to be associative if I did not make mistakes in my computation.
Thus I wonder whether someone has looked into the structure of this kind of sets, or, equivalently, the nonzero-scalar-multiplication-invariant property of (collection of) operators.
This set somehow reminds me of the Grassmannians, to which I know little.
Thanks!