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Czechoslovak Mathematical Journal, Vol. 71, No. 3, pp. 817-822, 2021

On a Kleinecke-Shirokov theorem

Vasile Lauric

Received March 7, 2020.   Published online March 8, 2021.
Abstract:  We prove that for normal operators $N_1, N_2\in\mathcal{L(H)},$ the generalized commutator $[N_1, N_2; X]$ approaches zero when $[N_1,N_2; [N_1, N_2; X]]$ tends to zero in the norm of the Schatten-von Neumann class $\mathcal{C}_p$ with $p>1$ and $X$ varies in a bounded set of such a class.
Keywords:  Kleinecke-Shirokov theorem; generalized commutator
Classification MSC:  47B47, 47B10, 47B20
DOI:  10.21136/CMJ.2021.0103-20
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References:
[1] A. Abdessemed, E. B. Davies: Some commutator estimates in the Schatten classes. J. Lond. Math. Soc., II. Ser. 39 (1989), 299-308. DOI 10.1112/jlms/s2-39.2.299 | MR 0991663 | Zbl 0692.47009
[2] S. T. M. Ackermans, S. J. L. van Eijndhoven, F. J. L. Martens: On almost commuting operators. Indag. Math. 45 (1983), 385-391. DOI 10.1016/S1385-7258(83)80015-8 | MR 0731821 | Zbl 0573.47024
[3] D. C. Kleinecke: On operator commutators. Proc. Am. Math. Soc. 8 (1957), 535-536. DOI 10.1090/S0002-9939-1957-0087914-4 | MR 0087914 | Zbl 0079.12904
[4] F. V. Shirokov: Proof of a conjecutre of Kaplansky. Usp. Mat. Nauk 11 (1956), 167-168. (In Russian.) MR 0087913 | Zbl 0070.34201
[5] V. Shulman: Some remarks on the Fuglede-Weiss theorem. Bull. Lond. Math. Soc. 28 (1996), 385-392. DOI 10.1112/blms/28.4.385 | MR 1384827 | Zbl 0892.47007
[6] V. Shulman, L. Turowska: Operator synthesis. II: Individual synthesis and linear operator equations. J. Reine Angew. Math. 590 (2006), 143-187. DOI 10.1515/CRELLE.2006.007 | MR 2208132 | Zbl 1094.47054
Affiliations:   Vasile Lauric, Department of Mathematics, Florida A& M University, 1601 S. Martin L. King Jr. Blvd., Tallahassee, FL 32307, USA, e-mail: vlauric@netzero.com
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