  We discuss transfer-function realization for multivariable holomorphic
functions mapping the unit polydisk or the right polyhalfplane into the
operator analogue of either the unit disk or the right halfplane
(Schur/Herglotz functions over either the unit polydisk or the right
polyhalfplane) which satisfy the appropriate stronger contractive/positive real
part condition for the values of these functions on commutative tuples of
strict contractions/strictly accretive operators (Schur--Agler/Herglotz--Agler
functions over either the unit polydisk or the right polyhalfplane). As
originally shown by Agler, the first case (polydisk to disk) can be solved via
unitary extensions of a partially defined isometry constructed in a canonical
way from a kernel decomposition for the function (the {\em lurking-isometry
method}). We show how a geometric reformulation of the lurking-isometry method
(embedding of a given isotropic subspace of a Kre\u{\i}n space into a
Lagrangian subspace---the {\em lurking-isotropic-subspace method}) can be used
to handle the second two cases (polydisk to halfplane and polyhalfplane to
disk), as well as the last case (polyhalfplane to halfplane) if an additional
growth condition at $\infty$ is imposed. For the general fourth case, we show
how a linear-fractional-transformation change of variable can be used to arrive
at the appropriate symmetrized nonhomogeneous Bessmertny\u{\i} long-resolvent
realization. We also indicate how this last result recovers the classical
integral representation formula for scalar-valued holomorphic functions mapping
the right halfplane into itself.

Functional Analysis