On Howland time-independent formulation of CP-divisible quantum evolutions

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.