Consider a closed connected Riemannian manifold $M$, together with the associated Hilbert space $L^2(M)$ defined with respect to the Riemannian volume density. Let $-\Delta$ be the positive Laplacian $H^2(M)\to L^2(M)$, and let $\{E_j\}_{j\in \mathbb N}$ be the sequence of its eigenvalues, ordered in a non-decreasing way. As the whole space $L^2(M)$ is complicated to study, we can consider instead a sequence of Hilbert spaces $\{V_j\}_{j\in \mathbb N}\subset L^2(M)$ in the limit $j\to \infty$, and try to find out how much this sequence of vector spaces (or an object related to it) tells us about the geometry of $M$.

$\{V_j\}_{j\in \mathbb N}$ could be a filtration of (a dense subspace of) $L^2(M)$, but it does not need to be one. Clearly, Weyl type spectral asymptotics correspond to the choice
$$
V_j:=\bigcup_{j'\leq j}\text{Eig}(\Delta,E_{j'}),
$$
where $\text{Eig}(\Delta,E_{j'})$ denotes the eigenspace of $\Delta$ corresponding to the eigenvalue $E_{j'}$. My question is: *Which other well-known examples of choices for $\{V_j\}_{j\in \mathbb N}$ are there?*

Actually, I am new to spectral geometry, and I do not even know what can be said about $M$ geometrically if we know *everything* about $L^2(M)$, that is I do not know a source which considers the question whether two Riemannian manifolds with isomorphic $L^2$-Hilbert spaces (such that the constant functions are respected) are isometric, diffeomorphic, homeomorphic, homotopy equivalent, etc. In the setting above, this corresponds to the trivial case of choosing $V_j=L^2(M)$ for every $j$. Surely, there must be tons of literature about these questions.

Thank you for your comments and answers.