  A realization or linearization is a triple, $(A,b,c)$, consisting of a
$d-$tuple, $A= (A _1, \cdots, A_d )$, $d\in \mathbb{N}$, of bounded linear
operators on a separable, complex Hilbert space, $\mathcal{H}$, and vectors
$b,c \in \mathcal{H}$. Any such realization defines a (uniformly) analytic
non-commutative (NC) function in an open neighbourhood of the origin, $0:= (0,
\cdots , 0)$, of the NC universe of $d-$tuples of square matrices of any fixed
size via the formula $h(X) = I \otimes b^* ( I \otimes I _{\mathcal{H}} - \sum
X_j \otimes A_j ) ^{-1} I \otimes c$.
  It is well-known that an NC function has a finite-dimensional realization if
and only if it is a non-commutative rational function that is defined at $0$.
Such finite realizations contain valuable information about the NC rational
functions they generate. By considering more general, infinite-dimensional
realizations we study, construct and characterize more general classes of
uniformly analytic NC functions. In particular, we show that an NC function,
$h$, is (uniformly) entire, if and only if it has a jointly compact and
quasinilpotent realization. Restricting our results to one variable shows that
an analytic Taylor-MacLaurin series extends globally to an entire or
meromorphic function if and only if it has a realization whose component
operator is compact and quasinilpotent, or compact, respectively. This then
motivates our definition of the set of global uniformly meromorphic NC
functions as the (universal) skew field (of fractions) generated by NC rational
expressions in the (semi-free ideal) ring of NC functions with jointly compact
realizations.
