Interplay of Simple and Selfadjoint-Ideal Semigroups in B(H)

TL;DR

This paper characterizes selfadjoint-ideal semigroups generated by operators in B(H), revealing their invariance under unitary transformations and exploring conditions under which they are simple or not, especially for normal, rank one, and selfadjoint operators.

Contribution

It introduces a new unitary invariant for B(H)-semigroups and provides a detailed characterization of SI semigroups generated by specific classes of operators, highlighting differences between SI and simplicity.

Findings

SI property is a unitary invariant for B(H)-semigroups.

Characterization of SI semigroups generated by normal and rank one operators.

Examples showing SI semigroups may or may not be simple, especially for selfadjoint generators.

Abstract

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.