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We prove that whenever the coherent information of a one-mode Gaussianchannel is non-zero its supremum is achieved for the infinite input power. Thisis a well established fact for the zero added classical noise, whereas thenonzero case has not been studied in detail. The presented analysis fills thegap for three canonical classes of one-mode Gaussian channels: the lossy,amplifying and additive classical noise channel class. For the remainingone-mode Gaussian channel classes the coherent information is known to vanish.

Coherent information of one-mode Gaussian channels -- the general case of nonzero added classical noise