  The additivity of the minimal output entropy and that of the $\chi$-capacity
are known to be equivalent for finite-dimensional irreducibly covariant
channels. In this paper we formulate a list of conditions allowing to establish
similar equivalence for infinite-dimensional covariant channels with
constrained input. This is then applied to Bosonic Gaussian channels with
quadratic input constraint to extend the classical capacity results of the
recent paper \cite{ghg} to the case where the complex structures associated
with the channel and with the constraint operator need not commute. In
particular, this implies a multimode generalization of the "threshold
condition", obtained for single mode in \cite{schaefer}) and the proof of the
fact that under this condition the classical "Gaussian capacity" resulting from
optimization over Gaussian inputs is equal to the full classical capacity. We
also investigate implications of the gauge-covariance condition for the single-
and multimode Bosonic Gaussian channels.
