Do operators that aren't endomorphism play a part in quantum? I know that given a quantum system with Hilbert space $\mathcal{H}$, Hermitian operators $A: \mathcal{H} \to \mathcal{H}$ correspond to observations.
This is a slightly vague question, but if we have two quantum systems $\mathcal{H_1}, \mathcal{H_2}$ (I'm imagining $L^2(X), L^2(Y)$), do operators $$A: \mathcal{H_1} \to \mathcal{H_2}$$ which have good properties (say unitarity, or the right analog of Hermiticity, which probably means you can break each of the spaces into orthogonal parts, where the map acts separately on each; this is probably like preserving orthogonal vectors) have physical significance?
A very rough guess: suppose I have a very violent experiment that converts a particle in one system to one in another system, then can it be described as such an operator?
 A: There are plenty of processes in quantum mechanics that can be loosely interpreted as "losing part of the system", or at least information about the system. These processes are generally described by operators that aren't endomorphisms, not strictly between the Hilbert spaces but between density matrices (i.e. trace 1 self adjoint endomorphisms on the Hilbert spaces).
Let $\mathcal H_{A,B}$ be Hilbert spaces and let $\mathcal D_{A,B}$ be the respective spaces of density matrices.
A regular old unitary evolution is the map
$$\mathcal U:\mathcal D_{AB}\to\mathcal D_{AB} \\\rho\mapsto U\rho U^\dagger$$
this is an endomorphism, the most elementary example of a map that is physically sensible and is not an endomorphism is the partial trace
$$ \mathrm{Tr}_B:\mathcal D_{AB}\to\mathcal D_A\\ \rho\mapsto\sum_{b}\langle b|\rho|b\rangle$$
where the $|b\rangle$ are a basis of $\mathcal H_B$. This just means that we had a system composed of $A$ and $B$, say an electron and a photon, but now we lost the photon. If we want to continue working with the electron, we must average over all the possible states the photon might have been in to model what whe know about the electron. This is what the partial trace does.
But what if something happens before we lose part of the system? Suppose I'm working with an electron, suddenly a photon passes by, it hits my electron and flies away. What will I observe? I need something like
$$\textrm{electron}\to\textrm{electron+photon}\to\textrm{interaction}\to \textrm{electron} $$
This is where another example of non endomorphism kicks in. The electron-photon system is isolated by itself, so it evolves unitarily, but if initially there was no photon in my system, I need some unitary evolution that makes the photon "appear", i.e. I need to map my "electron only" system to an "electron and photon" system. This can be done via an isometry.
$$ V:\mathcal H_A\to\mathcal H_{AB} : V^\dagger V=\mathbb 1$$
this is basically a unitary that "inflates" the system, it maps orthogonal vectors to orthogonal vectors, but since the dimension of the target space is larger than the one of the domain, it can't be surjective. This is simply equivalent to coupling your system with a "blank" state on $H_B$ and then unitarily evolving the whole thing, i.e. there are $|0\rangle_B$ and $U$ a unitary such that
$$V|\psi\rangle_A=U|\psi\rangle_A \otimes |0\rangle_B$$
So we took our electron, added a photon to the picture, and evolved them together modelling some interaction, by simply writing down
$$ \rho_A\mapsto V\rho_{A}V^\dagger$$
we now need to lose the photon, this is done with the partial trace
$$ \rho_A\mapsto \mathrm{Tr}_B(V\rho_AV^\dagger).$$
It turns out that any physically meaningful evolution can be written in this way, maps of these forms are called quantum channels, and this fact is known as Stinespring dilation theorem.
TL;DR: non endomorphisms are relevant when you want to model losing information about the system. Any physical evolution can be modelled as a unitary evolution followed by a loss of information.
