Two commuting operators associated with a tridiagonal pair

Let K denote a field and let V denote a vector space over K with finite positive dimension. We consider an ordered pair of linear transformations A:V→V and A∗:V→V that satisfy the following four conditions: (i) Each of A,A∗ is diagonalizable; (ii) there exists an ordering of the eigenspaces of A such that A∗Vi⊆Vi-1+Vi+Vi+1 for , where V-1=0 and Vd+1=0; (iii) there exists an ordering of the eigenspaces of A∗ such that for , where and ; (iv) there does not exist a subspace W of V such that AW⊆W, A∗W⊆W, W≠0, W≠V. We call such a pair a tridiagonal pair on V. It is known that d=δ; to avoid trivialities assume d⩾1. We show that there exists a unique linear transformation Δ:V→V such that and Δ(Vi+Vi+1+⋯+Vd)=V0+V1+⋯+Vd-i for . We show that there exists a unique linear transformation Ψ:V→V such that ΨVi⊆Vi-1+Vi+Vi+1 and for0⩽i⩽d, where Λ=(Δ-I)(θ0-θd)-1 and θ0 (resp. θd) denotes the eigenvalue of A associated with V0 (resp. Vd). We characterize Δ,Ψ in several ways. There are two well-known decompositions of V called the first and second split decomposition. We discuss how Δ,Ψ act on these decompositions. We also show how Δ,Ψ relate to each other. Along this line we have two main results. Our first main result is that Δ,Ψ commute. In the literature on TD pairs, there is a scalar β used to describe the eigenvalues. Our second main result is that each of Δ±1 is a polynomial of degree d in Ψ, under a minor assumption on β.