The title is borrowed from a 1969 paper by Paul Halmos. Two subspaces and of a Hilbert space are said to be in generic position if all four intersections , , , are trivial. It may be easier to visualize the condition by writing it as . The term “generic position” is due to Halmos, but the concept was considered before: e.g., in 1948 Dixmier called it “position p”.
Let us consider the finite-dimensional case: is either or . The dimension count shows that there are no pairs in generic position unless
- , and
Assume 1 and 2 from now on.
In the simplest case the situation is perfectly clear: two lines are in generic position if the angle between them is different from and . Any such pair of lines is equivalent to the pair of graphs up to rotation. Halmos proved that the same holds in general: there exists a decomposition and a linear operator such that the generic pair of subspaces if unitarily equivalent to .
If we have a preferred orthonormal basis in , it is natural to pay particular attention to coordinate subspaces, which are spanned by some subset of . Given a subspace , can we find a coordinate subspace such that and are in generic position? The answer is trivially no if contains some basis vector. When this is the only obstruction, as is easy to see:
In higher dimensions… later