Burnside's theorem: Irreducible pairs of transformations (Q1826827)

A family \({\mathcal F}\) of linear transformations acting on a finite \(n\)-dimensional complex vector space \(H\) is irreducible if they have no common nontrivial invariant subspaces. By Burnside's theorem all the words with factors (letters) belonging to such an \({\mathcal F}\) span the entire algebra \({\mathcal B}(H)\). A result of Radjavi suggests that the length of these words can be bounded and in case of two transformations \(A\) and \(B\) by \(n^2- 1\) [cf. \textit{H. Radjavi} and \textit{P. Rosenthal}, Simultaneous triangularization (2000; Zbl 0981.15007)]. Denote by \(msl(A,B)\) the minimum spanning length of the pair \((A,B)\). The full determination of \(msl(A,B)\) can be therefore regarded as a refinement of Burnside's theorem, but the general case remains open. The author describes \(msl(A,B)\) in the low-dimensional cases and if \(\dim H= n\geq 2\) then obtains that \(msl(A,B)\) is (i) equal to \(2n- 2\) when \(A\), \(B\), \(AB\), \(BA\) are linearly dependent and (ii) at most \(2n- 2\) in case that at least one of \(A\) and \(B\) is unicellular. It is interesting as to whether an example of \(msl(A,B)\) greater than \(2n- 2\) exists or not.