On extension of quantum channels and operations to the space of relatively bounded operators

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.