I was following Quantum Markov Processes from the book The Theory of Open Quantum Systems by Breuer and Petruccione. In the section The Markovian Quantum Master Equation they proceeds to 'construct the most general form for the generator $\mathcal{L}$ of a quantum dynamical semigroup.' Then I observed that almost the same approach is taken in the book Quantum Dynamical Semigroups and Applications by Alicki and Lendi, so I moved there.
We have the definition of $\mathcal{L}$: $$\frac{d}{dt}\rho_t=\mathcal{L}\rho_t$$ with $\rho_t=\Lambda_t\rho$. And this gives $\Lambda_t=e^{\mathcal{L}t}, t\ge0.$
For unbounded $\mathcal{L}$ they have used the following definition of exponential - $$e^{\mathcal{L}t}\rho=\lim\limits_{n \to \infty}\left(1-\frac{t}{n}\mathcal{L}\right)^{-n}\rho.$$ After that,
We find now a general form of $\mathcal{L}$ in the case of finite-dimensional Hilbert space $\mathcal{H_S}$ (dim $\mathcal{H_S} = N$). Introducing a linear basis {$F_k$}, $k =$ $0,1,... ,N^2 − 1$ in $\mathcal{B(H_S)}$ such that $F_0 = \mathbb{1}$ we may write a time-dependent version of $\Lambda\rho=\sum_{\alpha} W_{\alpha}\rho W_{\alpha}^*$ as follows, $$\Lambda_t\rho=\sum_{k,l=0}^{N^2-1}C_{kl}(t)F_k\rho F_l^*,$$ where $C_{kl}(t)$ is a positive-definite matrix.
The part after this is what I'm totally in the dark with. Let me paste the photo as it will be too much effort to type:
How are we getting this expression for $\mathcal{L}\rho$ with all these limits, $\epsilon$'s and all that? What I got is the following: $$\mathcal{L}\rho=\frac{d}{dt}\rho_t=\frac{d}{dt}{[\Lambda_t\rho]}=\frac{d}{dt}\left(\sum_{k,l=0}^{N^2-1}C_{kl}(t)F_k\rho F_l^*\right)$$ but can proceed no further.