  We define the domain of a linear fractional transformation in a space of
operators and show that both the affine automorphisms and the compositions of
symmetries act transitively on these domains. Further, we show that Liouville's
theorem holds for domains of linear fractional transformations, and, with an
additional trace class condition, so does the Riemann removable singularities
theorem. We also show that every biholomorphic mapping of the operator domain
$I < Z^*Z$ is a linear isometry when the space of operators is a complex Jordan
subalgebra of ${\cal L}(H)$ with the removable singularity property and that
every biholomorphic mapping of the operator domain $I + Z_1^*Z_1 < Z_2^*Z_2$ is
a linear map obtained by multiplication on the left and right by J-unitary and
unitary operators, respectively.
  Readers interested only in the finite dimensional case may identify our
spaces of operators with spaces of square and rectangular matrices.
