  We extend the de Branges-Beurling theorem characterizing the shift-invariant
spaces boundedly contained in the Hardy space of square-summable power series
to the full Fock space over $\mathbb{C} ^d$. Here, the full Fock space is
identified as the \emph{Non-commutative (NC) Hardy Space} of square-summable
Taylor series in several non-commuting variables. We then proceed to study
lattice operations on NC kernels and operator-valued multipliers between
vector-valued Fock spaces. In particular, we demonstrate that the
operator-valued Fock space multipliers with common coefficient range space form
a bounded general lattice modulo a natural equivalence relation.
