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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.


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I could respond in several ways, you can pick what you prefer.

1) People looked at the geometry of the unit sphere in Banach spaces, and more specifically in Banach algebras. For example, Geometrical Properties of the Unit Sphere of Banach Algebras by H. F. Bohnenblust and S. Karlin. Also V. D. Milman, "Geometry of Banach spaces II, geometry of the unit sphere", Russian Mathematical Surveys 26 (1971), 79–163.

2) The property of invariance under scalar multiplication is one of requirements of an algebra. To use the example you mentioned: the operators that leave subspace $Y$ invariant form an algebra. The invariance under scalar multiplication is already accounted for in this statement.

3) Consider two sets: $\mathcal A=\{T\colon \ell Te\ge 0\}$ and $\mathcal A'=\{T/\|T\|\colon \ell Te\ge 0\}$. Both contain the same amount of information, in the sense that you can get one from the other at once, without knowing $\ell$ and $e$. But $\mathcal A$ is closed under addition while $\mathcal A'$ is not. It looks like you lose something and gain nothing by turning attention to $\mathcal A'$.

4) If in addition to scalar multiplication, the property is also preserved under addition of operators and under limits, then you are dealing with an operator space. An operator space can be defined as a closed linear subspace of $B(\mathcal H)$. A good place to learn this topic is Introduction to Operator Theory by Pisier.

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