Consider a Q-matrix on a countable state space. (A Q-matrix is a matrix whose rows sum up to $0$, with nonpositive finite diagonal entries and nonnegative offdiagonal entries).
As explained for example in the book of Norris on Markov chains, every Q-matrix (without any further assumptions) defines a transition function (given by the minimal nonnegative solution of the corresponding backward equation).
Is the the semigroup associated to this minimal transition fucntion always Feller? (definition of Feller: functions vanishing at infinity are maped into functions vanishing at infinty).
I know that in general, not every continuous time Markov chain is Feller, but I guess that for this minimal chains it is always true without further assumptions on $Q$. Though I didn't find a statement like this in the books I consulted.
Has anybody a reference/proof/counterexample?