2018/06/04 by Sasmita Patnaik, Patnaik, Sasmita, Gary Weiss +1
Mathematics · #47B99 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #math.FA #math.OA #msc:47B99
paper · pdf · doi:10.48550/arxiv.1806.01272
26 Pages
arxiv created 2019/04/02 · arxiv updated 2019/04/03
This paper investigates a question of Radjavi: Which multiplicative semigroups in B(H) have all their ideals selfadjoint (called herein selfadjoint-ideal (SI) semigroups)? We proved this property is a unitary invariant for B(H)-semigroups, which invariant we believe is new. We characterize those SI semigroups S singly generated by T, for T a normal operator and for T a rank one operator. When T is nonselfadjoint and normal or rank one: S is an SI semigroup if and only if it is simple, except in one special rank one partial isometry case when our characterization yields S that are SI but not simple. So SI and simplicity are not equivalent notions. When T is selfadjoint, it is straightforward to see that S is always an SI semigroup, but we prove by examples they may or may not be simple, but for this case we do not have a characterization. The study of SI semigroups involves solving certain operator equations in the semigroups. A central theme of this paper is to study when and when not SI is equivalent to simple.