  The coding theorem for the entanglement-assisted communication via
infinite-dimensional quantum channel with linear constraint is extended to a
natural degree of generality. Relations between the entanglement-assisted
classical capacity and the $\chi$-capacity of constrained channels are obtained
and conditions for their coincidence are given. Sufficient conditions for
continuity of the entanglement-assisted classical capacity as a function of a
channel are obtained. Some applications of the obtained results to analysis of
Gaussian channels are considered.
  A general (continuous) version of the fundamental relation between the
coherent information and the measure of privacy of classical information
transmission by infinite-dimensional quantum channel is proved.
