  The classical capacity of phase-invariant Gaussian channels has been recently
determined under the assumption that such channels are memoryless. In this work
we generalize this result by deriving the classical capacity of a model of
quantum memory channel, in which the output states depend on the previous input
states. In particular we extend the analysis of [C. Lupo, et al., PRL and PRA
(2010)] from quantum limited channels to thermal attenuators and thermal
amplifiers. Our result applies in many situations in which the physical
communication channel is affected by nonzero memory and by thermal noise.
