This paper presents a linear dynamical operator described in terms of a\nrational transfer function, endowed with a well-defined and efficient\nback-propagation behavior for automatic derivatives computation. The operator\nenables end-to-end training of structured networks containing linear transfer\nfunctions and other differentiable units {by} exploiting standard deep learning\nsoftware.\n Two relevant applications of the operator in system identification are\npresented. The first one consists in the integration of {prediction error\nmethods} in deep learning. The dynamical operator is included as {the} last\nlayer of a neural network in order to obtain the optimal one-step-ahead\nprediction error.\n The second one considers identification of general block-oriented models from\nquantized data. These block-oriented models are constructed by combining linear\ndynamical operators with static nonlinearities described as standard\nfeed-forward neural networks. A custom loss function corresponding to the\nlog-likelihood of quantized output observations is defined. For gradient-based\noptimization, the derivatives of the log-likelihood are computed by applying\nthe back-propagation algorithm through the whole network. Two system\nidentification benchmarks are used to show the effectiveness of the proposed\nmethodologies.\n