J. Anderson [J. Reine Angew. Math., 291, 128-132 (1977; Zbl 0341.47025)] proved, as a special case of a more general theorem, that a rank one projection \(P\) on an infinite-dimensional Hilbert space can be written as a commutator, \(P= [S,T]= ST- TS\), where \(S\) and \(T\) can be taken in the Schatten-von Neumann class \({\mathcal C}_ p\) for any \(p> 4\). This raises the question whether \(p=4\) is possible. The answer is negative, as follows from: If \(S\in {\mathcal C}_ p\) and \(T\in {\mathcal C}_ q\), where \({1\over p}+{1\over q}\geq {1\over 2}\), and if \([S,T]\) is of finite rank, then trace \(([S,T])= 0\).
The theorem shows that some of the more general estimates in Anderson’s results also are best possible. A couple of related open questions are briefly discussed.