On the commuting properties of Taylor's spectrum (Q1209036)

For two operators \(A\) and \(B\) on a Banach space \(E\), it is well known that the nonzero spectrum of \(AB\) coincides with the nonzero spectrum of \(BA\). The author obtains a generalization of this statement for a pair of commuting \(n\)-tuples of operators \((A_ 1,\dots,A_ n)\) and \((B_ 1,\dots,B_ n)\). The main result is: if \((A_ 1,\dots,A_ n)\) and \((B_ 1,\dots,B_ n)\) crisscross commute (i.e., \(A_ i B_ j A_ k=A_ k B_ j A_ i\) and \(B_ i A_ k B_ j=B_ j A_ k B_ i\) for all triples of indices \(i\), \(j\), \(k\)), then the Taylor spectrum of the tuple \((A_ 1 B_ 1,\dots,A_ n B_ n)\) coincides with the Taylor spectrum of the tuple \((B_ 1 A_ 1,\dots,B_ n A_ n)\), except possibly for the point \((0,\dots,0)\).