While thinking about what it means for observables to be simultaneously measurable in quantum mechanics I came up with the following concepts, which I will call "linearly indexed" versions of standard linear algebra concepts.

- A $W$-indexed linear map $L: V \to V$ is a linear map $W \otimes V \to W \otimes V$.
- A $W$-indexed eigenspace for $L$ is a subspace $V'$ of $V$ such that $L = \lambda \otimes I$ on $W \otimes V'$.
- $\lambda : W \to W$ is called the $W$-indexed eigenvalue of $L$ for the $W$-indexed eigenspace $V'$.

When the spaces are over $\mathbb{C}$, say, and $W = \mathbb{C}$, we just recover the normal definition of eigenspace and eigenvalue $\lambda \in \mathbb{C}$.

It seems to me that these concepts allow us to talk about multidimensional observables of a quantum mechanical system in a direct way. An observable on a space of states $V$ is a Hermitian $W$-indexed linear map $V \to V$, and the values it can take are its $W$-indexed eigenvalues, which should to be Hermitian operators $W \to W$.

Is this something that is known and studied, or is this a red herring?