I am studying semi-groups and infinitesimal generators from the book by A. Pazy, and I wish to understand the semigroup and infinitesimal generator associated with a finite-state continuous time Markov chain, say $X = \{X_t\}_{t \geq 0}$, with state space $\chi,\ |\chi| = k$.\
I have read in some lecture notes that the $C_0$-semigroup $T = \{T_t\}_{t \geq 0}$, acting on the space of bounded linear operators on V $(BL(V))$, where $V = L^{\infty}\left(\chi \to \mathbb{R}\right)$, given by $T(t)f(x) = E[f(X_t)\vert X_0 = x]$ is the semigroup for the Markov process $X$. In the accompanying proof, its infinitesimal generator of $X$ is found to be its own transition rate matrix / Q-matrix $\Lambda = \left(\lambda_{ij}\right)_{k \times k}$.\
I do not understand this - how is $\Lambda$ an operator on $V$? By definition, $\Lambda$ can operate only on vectors with dimension $k \times 1$, but elements of $V$ are scalars. Is $V$ actually supposed to contain all possible outputs of $ L^{\infty}\left(\chi \to \mathbb{R}\right)$, because then the dimensions match.