On almost commuting Hermitian operators (Q806006)

It is an open problem if a pair of compact hermitian operators with ``small'' commutator can be ``well'' approximated by a commuting pair of hermitian operators. More precisely, it is not known if for every \(\epsilon >0\) there exists \(\delta >0\) such that whenever A, B are norm one, compact hermitian operators with \(\| [A,B]\| <\delta\), then one can find compact hermitian operators S, T satisfying \(\| S-A\| <\epsilon\), \(\| T-B\| <\epsilon\) and \([S,T]=0.\) The main result of this paper is that there exists a \(C>0\) such that for A, B of rank at most n the problem canbe solved with \(\delta =C\epsilon^{13/2}n^{-1/2}\). This gives an asymptotic improvement of an analogous result of \textit{C. Pearcy} and \textit{A. Shields} [J. Funct. Anal. 33, 332-338 (1979; Zbl 0425.15003)]. Obviously a complete answer of the problem would require \(\delta\) to be independent of n, but the present result shows that a counterexample, if it exists, must be of different nature than the counterexamples constructured by \textit{D. Voiculescu} [Acta Sci. Math. 45, 429-431 (1983; Zbl 0538.47003)] and \textit{M.-D. Choi} [Proc. Amer. Math. Soc. 102, No.3, 529-533 (1988; Zbl 0649.15005)] in order to show that the analogues of the problem for unitary and arbitrary operators, respectively, have a negative answer.