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On hermitian operators X and Y meeting the condition −Y ≤ X ≤ Y

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Abstract

We obtain a description of all pairs of Hermitian operators X and Y, which satisfy the condition −Y ≤ X ≤ Y. We give the examples of such operator pairs. Each of the presented examples leads us to the new weak majorization for the Hermitian operator pair. It is shown that this inequality does not necessarily imply the inequality |X| ≤ ZY Z* for any operator Z, ‖Z‖ ≤ 1. We prove that invertibility of Y follows from invertibility of operators X and A* A for Hermitian operators X and Y, Y ≥ 0 and an arbitrary operator A such that −AY A* ≤ X ≤ AY A*. We discuss one analog of triangle inequality found by the author in one earlier paper for pairs of Hermitian operators.

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References

  1. C. A. Akemann, J. Anderson, and G. K. Pedersen, Linear Multilinear Algebra 11(2), 167 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  2. A. M. Bikchentaev, Math. Notes 87(5–6), 768 (2010); translation fromMat. Zametki 87 (5), 787 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  3. A. M. Bikchentaev, An Inequality for Hermitian Operators, in Proceedings of Conference dedicated to 100th birthday of B. M. Gagaev “Algebra and Analysis” (Kazansk. Matem. Ob-vo, Kazan, 1997), p. 35.

    Google Scholar 

  4. A. M. Bikchentaev, Lobachevskii J. Math. 33(3), 216 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  5. A. M. Bikchentaev, Russ. Math. 56(2), 75 (2012); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2, 86 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  6. R. Bhatia, Matrix Analysis (Springer, 1997).

    Book  Google Scholar 

  7. J.-C. Bourin, O. Hirzallah, and F. Kittaneh, Linear Multilinear Algebra, 58(5–6), 645 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  8. J.-C. Bourin and E.-Y. Lee, Bull. London Math. Soc. 44(6), 1085 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Dixmier, Les algebres d’opérateurs dans l’espace Hilbertien (algebres de von Neumann) (Gauthier-Villars, Paris, 1969).

    MATH  Google Scholar 

  10. F. Hansen, Math. Ann. 246(3), 249 (1980).

    Article  MATH  Google Scholar 

  11. A.N. Sherstnev, Methods of bilinear forms in non-commutative measure and integral theory. (Metody bilineinykh form v nekommutativnoi teorii mery i integrala), (Moscow, Fizmatlit, 2008) (in Russian).

    MATH  Google Scholar 

  12. S. Strâtilâ and L. Zsidó, Lectures on von Neumann algebras (Abacus Press, England, 1979).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Kazan Federal University, Kazan, Russia

    Airat M. Bikchentaev

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  1. Airat M. Bikchentaev
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Correspondence to Airat M. Bikchentaev.

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(Submitted by D. H. Mushtari)

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Bikchentaev, A.M. On hermitian operators X and Y meeting the condition −Y ≤ X ≤ Y . Lobachevskii J Math 34, 227–233 (2013). https://doi.org/10.1134/S1995080213030037

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  • DOI: https://doi.org/10.1134/S1995080213030037

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