Raising and lowering maps for tridiagonal pairs

Let V denote a nonzero finite-dimensional vector space. A tridiagonal pair on V is an ordered pair A, A* of maps in End(V) suchthat (i) each of A, A* is diagonalizable; (ii) there exists an ordering {Vi}(i = 0)^(d) of the eigenspaces of A such thatA*Vi ⊆ Vi − 1 + Vi + Vi + 1 (0 ≤ i ≤ d), where V-¸1 = 0 and Vd + 1 = 0; (iii) there exists an ordering {Vi*}_(i = 0)^(δ) of the eigenspaces of A* such that AVi* ⊆ Vi − 1*+ Vi* + Vi + 1* (0 ≤ i ≤ δ), where V-1* = 0 and Vδ + 1* = 0; (iv) there does not exist a subspace W ⊆ V such that W ≠ 0, W ≠ V, AW ⊆ W, A*W ⊆ W. Assume that A, A* is a tridiagonal pair on V. It is known that d = δ. For 0 ≤ i ≤ d let θi (resp. θi*) denote the eigenvalue of A (resp. A*) for Vi (resp. Vi*). By construction, there exist R,F,L \in End(V) such that A = R + F + L andRVi* ⊆ Vi + 1*, FVi* ⊆ Vi*, LVi* ⊆ Vi − 1* (0 ≤ i ≤ d). For 0 ≤ i ≤ d define Ui = (V₀* + V₁* + ⋯ + Vi*) ∩ (Vi + Vi+1 + ⋯ + Vd).It is known that the sum V=sum{i=0}d Ui is direct. By construction, there exists R, L \in End(V) such that ℛ = A − θiI and ℒ = A* − θi*I on Ui (0 ≤ i ≤ d). It is known that ℛUi ⊆ Ui+1 and ℒUi ⊆ Ui-1 (0 ≤ i ≤ d), where U-1 = 0 and Ud + 1 = 0. In this paper, our main goal is to describe how R, F, L, ℛ, ℒ are related. We also give some results concerning injectivity/surjectivity and R, L.