The study of multiplicative B(H)-semigroups with selfadjoint-ideals property (called SI semigroups, in short) is related to the solvability of a certain operator equation. This introduces a new unitary invariant property for B(H)-semigroups. We characterized singly generated SI semigroups for a normal operator and for a rank one operator. The SI property is closely related to the simplicity of B(H)-semigroups.