The motivations for writing the present monograph are three-fold: firstly from a physical point of view and secondly from two related, but different mathematical angles.

At the present time our mathematical understanding of a conservative quantum mechanical system is reasonably complete, both from the direction of a consistent abstract theory as well as from the one of mathematical theories of applications in many explicit physical systems like atoms, molecules etc. (see for example the books [12] and [108]). However, a nonconservative (open/dissipative) quantum mechanical system does not enjoy a similar status. Over the last seven decades there have been many attempts to make a theory of open quantum systems beginning with Pauli [104]. Some of the typical references are: Van Hove [126], Ford et al. [52], along with the mathematical monograph of Davies [35]. The physicists' Master equation (or Langevin equation) was believed to describe the evolution of a nonconservative open quantum (or classical) mechanical system, a mathematical description of which can be found in Feller's book [50].

Physically, one can conceive of an open system as the ‘smaller subsystem’ of a total ensemble in which the system is in interaction with its ‘larger’ environment (sometimes called the bath or reservoir). The total ensemble with a very large number of degrees of freedom undergoes (conservative) evolution, obeying the laws of standard quantum mechanics. However, for various reasons, practical or otherwise, it is of interest only to observe the system and not the reservoir, and this ‘reduced dynamics’ in a certain sense obeys the Master equation (for a more precise description of these, see [35]).