  We analyse possibility to extend a quantum operation (sub-unital normal CP
linear map on the algebra $B(H)$ of bounded operators on a separable Hilbert
space $H$) to the space of all operators on $H$ relatively bounded w.r.t. a
given positive unbounded operator.
  We show that a quantum operation $\,\Phi\,$ can be uniquely extended to a
bounded linear operator on the Banach space of all $\sqrt{G}$-bounded operators
on $H$ provided that the operation $\Phi$ is $G$-limited: the predual operation
$\Phi_*$ maps the set of positive trace class operators $\rho$ with finite
$\mathrm{Tr}\rho G$ into itself.
  Assuming that $G$ has discrete spectrum of finite multiplicity we prove that
for a wide class of quantum operations the existence of the above extension
implies the $G$-limited property.
  Applications to the theory of Bosonic Gaussian channels are considered.
