The mathematical theory of quantum feedback networks has recently been developed for general open quantum dynamical systems interacting with bosonic input fields. In this article we show, for the special case of linear dynamical systems Markovian systems with instantaneous feedback connections, that the transfer functions can be deduced and agree with the algebraic rules obtained in the nonlinear case. Using these rules, we derive the the transfer functions for linear quantum systems in series, in cascade, and in feedback arrangements mediated by beam splitter devices.

19 pages, Latex figures in text