Existence and uniqueness of \(C_0\)-semigroup in \(L^\infty\): A new topological approach
A sub-Markov semigroup in \(L^\infty\) is in general not strongly continuous with respect to the norm topology. The authors deal with this problem in the general setting of sequentially complete linear vector spaces \(X\). A variant of Phillips' theorem on dual semigroups is presented and several results on generators are given. As is well-known, if \(T(t)\) is a \(C_0\)-semigroup on a Banach space \(X\), then \(T^*(t)\) is not necessarily a \(C_0\)-semigroup on the dual \(X^*\) with the strong topology. Therefore the authors introduce a new topology on \(X^*\). By \(C(X^*,X)\) they denote the topology of uniform convergence on compact subsets of \(X\) and they prove the equivalence of the following properties: (a) \(T(t)\) is a \(C_0\)-semigroup on \((X\); norm topology); (b) \(T^*(t)\) is a \(C_0\)-semigroup on \((X^*;C(X^*,X))\); (c) \(\lim_{t \downarrow 0}T^*(t) y=y\) in the weak topology \(\sigma(X^*,X)\), \(\forall y\in X^*\). Since \(L^\infty= (L^1)^*\) they use \(C(L^\infty, L^1)\) as a natural topology for studying \(C_0\)-semigroups on \(L^\infty\). Thus the usual sub-Markov semigroups from the literature become \(C_0\)-semigroups. Moreover, by the Grothendieck theorem they prove that the space \((L^\infty; C(L^\infty,L^2))\) is complete and give a simplified Hille-Yosida theorem. Subsequently, the authors present several important pre-generators for which they can prove \(L^\infty\) uniqueness.
scientific article; zbMATH DE number 1754843