| Title: | Commutators greater than a perturbation of the identity |
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| Authors: | ID Drnovšek, Roman (Author)
ID Kandić, Marko (Author) |
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| Files: |  PDF - Presentation file, download (326,77 KB)
MD5: 611225E70FCA0B0EA2CB44B815D522F0
 URL - Source URL, visit https://www.sciencedirect.com/science/article/pii/S0022247X24006358
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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| Abstract: | 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. |
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| Keywords: | Banach lattices, positive operators, commutators, ordered normed algebras |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.01.2025 |
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| Year of publishing: | 2025 |
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| Number of pages: | 11 str. |
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| Numbering: | Vol. 541, iss. 2, [article no.] 128713 |
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| PID: | 20.500.12556/DiRROS-20458  |
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| UDC: | 517.983 |
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| ISSN on article: | 0022-247X |
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| DOI: | 10.1016/j.jmaa.2024.128713 |
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| COBISS.SI-ID: | 208157699 |
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| Note: |
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| Publication date in DiRROS: | 19.09.2024 |
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| Views: | 687 |
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| Downloads: | 405 |
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