Commutators greater than a perturbation of the identity

Drnovšek, Roman (author), Kandić, Marko (author)

Let $a$ and $b$ be elements of an ordered normed algebra ${\mathcal A}$ with unit $e$. Suppose that the element $a$ is positive and that for some $\varepsilon > 0$ there exists an element $x\in {\mathcal A}$ with $\|x\|\leq \varepsilon$ such that $ab-ba \geq e+x$. If the norm on ${\mathcal A}$ is monotone, then we show $\|a\|\cdot \|b\|\geq \tfrac{1}{2} \ln \tfrac{1}{\varepsilon}$, which can be viewed as an order analog of Popa's quantitative result for commutators of operators on Hilbert spaces. We also give a relevant example of positive operators $A$ and $B$ on the Hilbert lattice $\ell^2$ such that their commutator $A B - B A$ is greater than an arbitrarily small perturbation of the identity operator.

• Language: English
• Keywords: Banach lattices, positive operators, commutators, ordered normed algebras
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2025
• Number of pages: 11 str.
• Numbering: Vol. 541, iss. 2, art. 128713
• PID: 20.500.12556/RUL-162296
• UDC: 517.983
• ISSN on article: 0022-247X
• DOI: 10.1016/j.jmaa.2024.128713
• COBISS.SI-ID: 208157699
• Publication date in RUL: 24.09.2024

Record is a part of a journal

• Title: Journal of mathematical analysis and applications
• Shortened title: J. math. anal. appl.
• Publisher: Elsevier
• ISSN: 0022-247X
• COBISS.SI-ID: 3081231

Licences

• License: CC BY-NC-ND 4.0, Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International
• Link: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Description: The most restrictive Creative Commons license. This only allows people to download and share the work for no commercial gain and for no other purposes.

Projects

• Funder: ARIS - Slovenian Research and Innovation Agency
• Project number: P1-0222
• Name: Algebra, teorija operatorjev in finančna matematika

• Funder: ARIS - Slovenian Research and Innovation Agency
• Project number: N1-0217
• Name: Nekomutativna realna algebraična geometrija s sledjo