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

Sasmita Patnaik; Gary Weiss

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arXiv · 1806.01272

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

Sasmita Patnaik·Gary Weiss

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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.

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BibTeX RIS

Sasmita Patnaik, Gary Weiss. 2018-06-04. Interplay of Simple and Selfadjoint-Ideal Semigroups in B(H). https://arxiv.org/abs/1806.01272

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