  We extend Howland time-independent formalism to the case of completely
positive and trace preserving dynamics of finite dimensional open quantum
systems governed by periodic, time dependent Lindbladian in Weak Coupling
Limit, expanding our result from previous papers. We propose the Bochner space
of periodic, square integrable matrix valued functions, as well as its tensor
product representation, as the generalized space of states within the
time-independent formalism. We examine some densely defined operators on this
space, together with their Fourier-like expansions and address some problems
related to their convergence by employing general results on Banach-space
valued Fourier series, such as the generalized Carleson-Hunt theorem. We
formulate Markovian dynamics in the generalized space of states by constructing
appropriate time-independent Lindbladian in standard
(Lindblad-Gorini-Kossakowski-Sudarshan) form, as well as one-parameter
semigroup of bounded evolution maps. We show their similarity with Markovian
generators and dynamical maps defined on matrix space, i.e. the generator still
possesses a standard form (extended by closed perturbation) and the resulting
semigroup is also completely positive, trace preserving and a contraction.
