Which operators are the self-commutators of compact operators?

Author:

Fan Peng,Fong Che Kao

Abstract

It is shown that, for a hermitian operator T on a Hubert space H, the following three statements are equivalent: (i) T = A A A A T = {A^ \ast }A - A{A^ \ast } for some compact operator A, (ii) there is an orthonormal basis { b j } \{ {b_j}\} such that T b j , b j = 0 \langle T{b_j},{b_j}\rangle = 0 for all j, and (iii) tr T + = tr T {\text {tr}}\,{T^ + } = {\text {tr}}\,{T^ - } (possibly infinite) where T + = ( | T | + T ) / 2 {T^ + } = (|T| + T)/2 and T = ( | T | T ) / 2 {T^ - } = (|T| - T)/2 .

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference7 articles.

1. Commutators of operators on Hilbert space;Brown, Arlen;Canadian J. Math.,1965

2. P. Fillmore, C. K. Fong and A. Sourour, Real parts of quasi-nilpotent operators, Proc. Edinburgh Math. Soc. (to appear).

3. Translations of Mathematical Monographs, Vol. 18;Gohberg, I. C.,1969

4. C. Pearcy, Some unsolved problems in operator theory, preprint, 1972.

5. On commutators in ideals of compact operators;Pearcy, Carl;Michigan Math. J.,1971

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