There are several 'standard' notions of convergence of a sequence of operators $(A_n)_{n=1}^{\infty}$ on a given Hilbert space. Norm convergence, strong convergence, weak convergence, and also equivalent definitions for resolvent convergence. From what I know, all of these require that the Hilbert space is fixed and independent of $n$.
I wanted to ask whether there is also some notion of convergence when the Hilbert space is allowed to change in $n$ (so we have also a sequence of Hilbert spaces - $H_n$.
I have several examples in mind of cases where one would like to discuss such a convergence. Here is one which relatively simple to describe. Consider the metric space $$X_n=\{(x,0)\in \mathbb R^2 : x\in[0,1]\}\cup \left\{\left(\frac{1+n}{2n},y\right)\in \mathbb R^2 : y\in[0,1]\right\}$$ This is a sequence of lines with an additional line "sticking" out of them. This line moves with $n$ and approaches the center. So these are "almost" manifolds, up to one singular point.
One can consider the Laplacian $-\frac{d^2}{dx^2}$ acting on some dense subspace of $L^2(X_n)$ (with some appropriate boundary conditions - for instance continuity + Neumann condition at the gluing point). I would like to think that in "some sense", this sequence of operator converges to the Laplacian (with the same boundary conditions) defined on the "limiting space" $X$ which is the limit in the Hausdorff metric of $X_n$. Yet, none of the usual notions of convergence can be applied here, since the two operators are not "comparable" (you cannot subtract functions in the two different Hilbert spaces).
If the example above is unclear I can explain more. There are a lot of similar examples one can come up with, so pick your favorite. Did you ever come across a notion of convergence which might fit these sorts of examples? I am especially interested in being able to compare the spectra of these operators.
Thanks in advance!