  Differential equations on spaces of operators are very little developed in
Mathematics, being in general very challenging. Here, we study a novel system
of such (non-linear) differential equations. We show it has a unique solution
for all times, for instance in the operator or Hilbert-Schmidt norm topologies.
This system presents remarkable ellipticity properties that turn out to be
crucial for the study of the infinite-time limit of its solution, which is
proven under relatively weak, albeit probably not necessary, hypotheses on the
initial data. This system of differential equations is the elliptic counterpart
of an hyperbolic flow applied to quantum field theory to diagonalize
Hamiltonians that are quadratic in the bosonic field. In a similar way, this
elliptic flow, in particular its asymptotics, has application in quantum field
theory: it can be used to diagonalize Hamiltonians that are quadratic in the
fermionic field while giving new explicit expressions and properties of these
pivotal Hamiltonians of quantum field theory and quantum statistical mechanics.
