A Pearcy-Shields problem (Q1277439)

Let \(A\) be a linear selfadjoint operator, acting in a complex Hilbert space \(H\) and having a countable spectrum \(s(A)\) of the form \(s(A)=s_1\cup \ldots \cup s_N\), where the \(s_j\) are disjoint bounded compacts. Suppose that for any \(\varepsilon>0\) there exists a partition \(U\) of the spectrum such that \[ \sup_{1\leq j\leq N}\text{diam } s_j \leq 2\varepsilon,\quad d(\varepsilon, U)=\inf\text{dist} (s_i,s_j)>0 \quad (i\neq j). \] Let \(d(\varepsilon)=\sup_{U} d(\varepsilon, U),\) and let \(B\) be a linear operator which is subordinate to \(A.\) Under some conditions on \(A\) and \(B\) it is proved that if \[ \|AB-BA \|\leq 2d(\varepsilon)\varepsilon/\pi \] for some \(\varepsilon\), then there exists a selfadjoint operator \(A'\) and a linear operator \(B'\) such that \(\|A-A'\|\leq\varepsilon\), \(\|B-B'\|\leq\varepsilon\).