A theory of strongly continuous semigroups in terms of Lie generators
For a separable complete metric space \(X\) denote by \({\mathcal F} (X)\) the set of all continuous transformations of \(X\) into itself and by \({\mathcal C} (X)\) the linear space of all bounded continuous real-valued functions from \(X\). A function \(T: [0, \infty)\to {\mathcal F} (X)\) is called jointly continuous semigroup in \(X\) if \(T(0)\) is the identity, \(T(t) T(s)= T(t+ s)\) for all \(t,s \geq 0\), \(T(-)\) is continuous from \([0, \infty)\) to \(X\) for every \(x\in X\), and the mapping \((t, x)\mapsto T(t) x\) from \([0, \infty) \times X\) to \(X\) is jointly continuous. The Lie generator of \(T\) is the linear operator \(A\) in \({\mathcal C} (X)\) consisting of all pairs \((f, g)\) of functions such that \(g(x)= \lim_{t\leftarrow 0} {1\over t} [f(T (t)x)- f(x)]\) for any \(x\in X\). A necessary and sufficient condition for a linear operator in \({\mathcal C} (X)\) to be a Lie generator of a jointly continuous semigroup is given. Examples and applications are discussed.
scientific article; zbMATH DE number 871739