Let $V$ be an $n$-dimensional vector space, and let $M\subset V\oplus V$ be an $n$-dimensional subspace. It is known that I can identify $M$ (non-uniquely) with a frame, i.e., with an injective linear operator $$\left( \begin{array}{c} X \\ Y \end{array} \right):V\rightarrow V\oplus V$$ whose range is $M$ (here, $X,Y$ are $n\times n$ matrices).
Now, assume that $M=M_1\oplus M_2$ where $M_1,M_2$ are some subspaces of $V\oplus V$ (no specific restrictions about dimension). Is there some natural way to express the frame associated with $M$ using the "smaller" subspaces $M_1, M_2$?
Using some elementary arguments I can show the following:
(so $M$ splits into the direct sum of restricted images to $M_1, M_2$). But I am looking for something which is possibly a bit stronger - I'd rather not compute the entire frame on $M$ (and then take the restriction), but to compute things only on the smaller subspaces if possible. Naively, $M_1, M_2$ don't have an associated frame (since their dimension is not $n$), but perhaps there is still some simple way to express the frame of $M$ using only information about $M_1, M_2$?
If it helps somehow, I am especially interested in the case where the subspace $M$ is Lagrangian with respect to the standard symplectic form. But feel free to ignore that if you wish and just consider it as a linear algebra problem.