20.500.12556/RUL-162296 Commutators greater than a perturbation of the identity 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. Banach lattices positive operators commutators ordered normed algebras true false true Angleški jezik Ni določen Članek v reviji 2024-09-21 05:10:36 2024-09-24 11:31:40 2024-09-25 04:18:32 0000-00-00 00:00:00 2025 0 0 11 str. iss. 2, art. 128713 Vol. 541 Jan. 2025 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 517.983 0022-247X 10.1016/j.jmaa.2024.128713 208157699 1-s2.0-S0022247X24006358-main.pdf 1-s2.0-S0022247X24006358-main.pdf 1 B1BBB58EE9984BCA941D994D496A8377 23d607bbb879e88765678d7ead7641aa25b74b3d9b695ff2a5fd5afe9dbfe0d3 cafa7f6d-7a57-11ef-b232-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=191832 https://www.sciencedirect.com/science/article/pii/S0022247X24006358 0 f76fbafe-77c6-11ef-b232-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=191502 Fakulteta za matematiko in fiziko 0 0 0