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On a certain type of commutators of operators

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Abstract

LetH be a separable infinite-dimensional Hilbert space and letC be a normal operator andG a compact operator onH. It is proved that the following four conditions are equivalent.

  1. 1.

    C +G is a commutatorAB-BA with self-adjointA.

  2. 2.

    There exists an infinite orthonormal sequencee j inH such that |Σ nj =1 (Ce j, ej)| is bounded.

  3. 3.

    C is not of the formC 1C 2 whereC 1 has finite dimensional domain andC 2 satisfies inf {|(C 2 x, x)|: ‖x‖=1}>0.

  4. 4.

    0 is in the convex hull of the set of limit points of spC.

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References

  1. A. Brown, P. R. Halmos and C. Pearcy,Commutators of operators on Hilbert space, Canad. J. Math.17 (1965), 695–708.

    MATH  MathSciNet  Google Scholar 

  2. M. David,On a certain type of commutator, J. Math. Mech.19 (1970), 665–680.

    MATH  MathSciNet  Google Scholar 

  3. H. Radjavi,Structure of A*A-AA*, J. Math. Mech.16 (1966), 19–26.

    MATH  MathSciNet  Google Scholar 

  4. F. Riesz and B. Sz.-Nagy,Functional Analysis, Frederick Ungar Publ. Co., N.Y., 1955.

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David, M. On a certain type of commutators of operators. Israel J. Math. 9, 34–42 (1971). https://doi.org/10.1007/BF02771617

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  • DOI: https://doi.org/10.1007/BF02771617

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