The split decomposition of a tridiagonal pair

Let K denote a field and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A : V → V and A∗ : V → V that satisfy (i)–(iv) below:(i)Each of A, A∗ is diagonalizable.(ii)There exists an ordering V0, V1, …, Vd of the eigenspaces of A such that A∗Vi⊆Vi-1+Vi+Vi+1 for 0 ⩽ i ⩽ d, where V-1 = 0, Vd+1 = 0.(iii)There exists an ordering of the eigenspaces of A∗ such that for 0 ⩽ i ⩽ δ, where , .(iv)There is no subspace W of V such that both AW⊆W,A∗W⊆W, other than W = 0 and W = V.We call such a pair a tridiagonal pair on V. In this note we obtain two results. First, we show that each of A, A∗ is determined up to affine transformation by the Vi and . Secondly, we characterize the case in which the Vi and all have dimension one. We prove both results using a certain decomposition of V called the split decomposition.