A Theory of Strongly Continuous Semigroups in Terms of Lie Generators

LetXdenote a complete separable metric space, and let C(X) denote the linear space of all bounded continuous real-valued functions onX. A semigroupTof transformations fromXintoXis said to be jointly continuous if the mapping (t, x)→T(t) xis jointly continuous from [0, ∞)×XintoX. The Lie generator of such a semigroupTis the linear operator in C(X) consisting of all ordered pairs (f, g) such thatf, g∈C(X), and for eachx∈X, g(x) is the derivative at 0 off(T(·) x). We completely characterize such Lie generators and establish the canonical exponential formula for the original semigroup in terms of powers of resolvents of its Lie generator. The only topological notions needed in the characterization are two notions of sequential convergence, pointwise and strict. A sequence in C(X) converges strictly if the sequence is uniformly bounded in the supremum norm and converges uniformly on compact subsets ofX.