Semigroups of operators on locally convex spaces

Abstract

Let X X be a complex Hausdorff locally convex topological linear space and L ( X ) L(X) the family of all continuous linear operators on X X . This paper discusses the generation and perturbation theory for C 0 {C_0} semigroups { S ( ξ ) : ξ ⩾ 0 } ⊂ L ( X ) \{ S(\xi ):\xi \geqslant 0\} \subset L(X) such that for each continuous seminorm p p on X X there exist a positive number σ p {\sigma _p} and a continuous seminorm q q on X X with p ( S ( ξ ) x ) ⩽ e σ p ξ q ( x ) p(S(\xi )x) \leqslant {e^{^\sigma {p^\xi }}}q(x) for all ξ ⩾ 0 \xi \geqslant 0 and x ∈ X x \in X . These semigroups are studied by means of a realization of X X as a projective limit of Banach spaces, using certain naturally-defined operators and C 0 {C_0} semigroups on these Banach spaces to connect the present results to the classical Hille-Yosida-Phillips theory.