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

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) for some compact operator *A*, (ii) there is an orthonormal basis such that for all *j*, and (iii) (possibly infinite) where and .

**[1]**Arlen Brown, P. R. Halmos, and Carl Pearcy,*Commutators of operators on Hilbert space*, Canad. J. Math.**17**(1965), 695–708. MR**0203460****[2]**P. Fillmore, C. K. Fong and A. Sourour,*Real parts of quasi-nilpotent operators*, Proc. Edinburgh Math. Soc. (to appear).**[3]**I. C. Gohberg and M. G. Kreĭn,*Introduction to the theory of linear nonselfadjoint operators*, Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. MR**0246142****[4]**C. Pearcy,*Some unsolved problems in operator theory*, preprint, 1972.**[5]**Carl Pearcy and David Topping,*On commutators in ideals of compact operators*, Michigan Math. J.**18**(1971), 247–252. MR**0284853****[6]**Heydar Radjavi,*Structure of 𝐴*𝐴-𝐴𝐴**, J. Math. Mech.**16**(1966), 19–26. MR**0203482****[7]**Joseph G. Stampfli,*Hyponormal operators*, Pacific J. Math.**12**(1962), 1453–1458. MR**0149282**

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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