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.
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1.
C +G is a commutatorAB-BA with self-adjointA.
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2.
There exists an infinite orthonormal sequencee j inH such that |Σ nj =1 (Ce j, ej)| is bounded.
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3.
C is not of the formC 1 ⊕C 2 whereC 1 has finite dimensional domain andC 2 satisfies inf {|(C 2 x, x)|: ‖x‖=1}>0.
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4.
0 is in the convex hull of the set of limit points of spC.
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References
A. Brown, P. R. Halmos and C. Pearcy,Commutators of operators on Hilbert space, Canad. J. Math.17 (1965), 695–708.
M. David,On a certain type of commutator, J. Math. Mech.19 (1970), 665–680.
H. Radjavi,Structure of A*A-AA*, J. Math. Mech.16 (1966), 19–26.
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