On the convergence of semi-groups of operators
Shinnosuke Ôharu · Proceedings of the Japan Academy Series A Mathematical Sciences · 1966
1. Let X be a locally convex, sequentially complete, linear topological space and { U, ().t>= 0}=, be a sequence of semi-groups of operators on X, satisfying the following conditions:(i) Uo('-I, U, (, U, ,/,,,'' t, t'>0,= (ii) lim U,(l x U,(2) x, ,o>=0, X, t-* (iii) {U(')} are equi-continuous in t and n, i.e., for any continuous semi-norm p on X, there exists a continuous semi-norm q on X, independent of t and n, such that p(U,(')x)<=q(x)x e X.And let F (') be the infinitesimal generator of {Ut(')}t_o i.e., F()x-lim h-( U( ") I)x.hOWe consider the following condition (A).(A) There exists a dense linear subset O(F()) such that lim (F()x F(')x)-O