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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Which operators are the self-commutators of compact operators?


Authors: Peng Fan and Che Kao Fong
Journal: Proc. Amer. Math. Soc. 80 (1980), 58-60
MSC: Primary 47B05; Secondary 47B47
DOI: https://doi.org/10.1090/S0002-9939-1980-0574508-X
MathSciNet review: 574508
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Abstract: It is shown that, for a hermitian operator T on a Hubert space H, the following three statements are equivalent: (i) $ T = {A^ \ast }A - A{A^ \ast }$ for some compact operator A, (ii) there is an orthonormal basis $ \{ {b_j}\} $ such that $ \langle T{b_j},{b_j}\rangle = 0$ for all j, and (iii) $ {\text{tr}}\,{T^ + } = {\text{tr}}\,{T^ - }$ (possibly infinite) where $ {T^ + } = (\vert T\vert + T)/2$ and $ {T^ - } = (\vert T\vert - T)/2$.


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DOI: https://doi.org/10.1090/S0002-9939-1980-0574508-X
Keywords: Self-commutator, compact operator, trace
Article copyright: © Copyright 1980 American Mathematical Society