Burnside's theorem: irreducible pairs of transformations

By Burnside's theorem, if the linear transformations A and B, acting on a finite-dimensional complex vector space H, have no common nontrivial invariant subspaces, the words in A and B span B(H). Call the minimum spanning length of the pair {A,B} the smallest positive integer l with the property that words in A and B of length at most l span B(H). Let msl(A,B) denote the minimum spanning length. If dim H=2, msl(A, B)=2 and if dint H=3, msl(A, B)=3 or 4. If dim Hgreater than or equal to4, msl(A, B)less than or equal ton(2)-3. If dim H=ngreater than or equal to2 then (i) msl(A,B)=2n-2 if (A,B, AB, BA) is linearly dependent, (ii) if B is unicellular. then msl(A,B)less than or equal to2n-2, where the inequality is sharp, and it can happen that msl (A,B)=n. (C) 2004 Elsevier Inc. All rights reserved.