On semigroups generated by topologically nilpotent elements
Lars Ingelstam · Illinois Journal of Mathematics · 1969
LARs ]NGELSTAM In a Banach algebra A one can define, for each element x, ixp x Z:---I x/n! or, if A has identity e, exp x e ixp x.Then, for a given x, a --, ixp ax (a -exp ax) is a one-parameter semigroup under circle composition (group under multiplication).In this paper we deal with the following question"How "bounded" can such a semigroup be when the generator x is "topologically very nilpotent"?In [4, pp.248, 259] the following results are given: I.If limn xn 1/ 0 and x 0, then ixp ax can not be bounded ia norm for all a.II.If lim n xn 1/n 0 and x 0, then ixp ax can not be bounded in norm for a > 0.This type of problem has arisen in two different contexts.In the paper by