  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.
