Infinitesimal Operators and the Distribution of Anticommutators and Commutators
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910450176753664 |
|---|---|
| author | Mingo, James A. Tseng, Pei-Lun |
| author_facet | Mingo, James A. Tseng, Pei-Lun |
| contents | In an infinitesimal probability space we consider operators which are infinitesimally free and one of which is infinitesimal, in that all its moments vanish. Many previously analysed random matrix models are captured by this framework. We show that there is a simple way of finding non-commutative distributions involving infinitesimal operators and apply this to the commutator and anticommutator. We show the joint infinitesimal distribution of an operator and an infinitesimal idempotent gives us the Boolean cumulants of the given operator. We also show that Boolean cumulants can be expressed as infinitesimal moments thus giving matrix models which exhibit asymptotic Boolean independence and monotone independence. Finally we demonstrate a connection to the Markov-Krein transform. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_02064 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Infinitesimal Operators and the Distribution of Anticommutators and Commutators Mingo, James A. Tseng, Pei-Lun Operator Algebras 46L54 In an infinitesimal probability space we consider operators which are infinitesimally free and one of which is infinitesimal, in that all its moments vanish. Many previously analysed random matrix models are captured by this framework. We show that there is a simple way of finding non-commutative distributions involving infinitesimal operators and apply this to the commutator and anticommutator. We show the joint infinitesimal distribution of an operator and an infinitesimal idempotent gives us the Boolean cumulants of the given operator. We also show that Boolean cumulants can be expressed as infinitesimal moments thus giving matrix models which exhibit asymptotic Boolean independence and monotone independence. Finally we demonstrate a connection to the Markov-Krein transform. |
| title | Infinitesimal Operators and the Distribution of Anticommutators and Commutators |
| topic | Operator Algebras 46L54 |
| url | https://arxiv.org/abs/2308.02064 |