The Transfer Matrix as Operator or Input/Output Map

In the standard discrete-time and continuous-time cases and for relevant classes of signals, we interpret the transfer matrix of an input/output behavior as transfer operator which maps input signals to output signals. In the most important settings, this action is given by convolution. The transfer operator generalizes the gain matrix. We characterize external stability and, in particular, bounded input/bounded output (BIBO) stability of transfer operators. The derived theory is then applied to electrical and mechanical networks.