Two commuting operators associated with a tridiagonal pair
Abstract
Let denote a field and let V denote a vector space over 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 Vii=0d of the eigenspaces of A such that A*Vi⊂eq Vi-1+Vi+Vi+1 for 0≤ i≤ d, where V-1=0 and Vd+1=0; (iii) there exists an ordering V*ii=0δ of the eigenspaces of A* such that AV*i⊂eq V*i-1+V*i+V*i+1 for 0≤ i≤δ, where V*-1=0 and V*δ+1=0; (iv) there does not exist a subspace W of V such that AW⊂eq W, A*W⊂eq W, W≠0, W≠ V. We call such a pair a TD pair on V. It is known that d=δ; to avoid trivialities assume d≥ 1. We show that there exists a unique linear transformation :V V such that ( -I)V*i⊂eq V*0+V*1+...+V*i-1 and (Vi+Vi+1+...+Vd)=V0 +V1+...+Vd-i for 0≤ i ≤ d. We show that there exists a unique linear transformation :V V such that Vi⊂eq Vi-1+Vi+Vi+1 and (-)V*i⊂eq V*0+V*1+...+V*i-2 for 0≤ i≤ d, where =(-I)(θ0-θd)-1 and θ0 (resp θd) denotes the eigenvalue of A associated with V0 (resp Vd). We characterize , in several ways. There are two well-known decompositions of V called the first and second split decomposition. We discuss how , act on these decompositions. We also show how , relate to each other. Along this line we have two main results. Our first main result is that , 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 1 is a polynomial of degree d in , under a minor assumption on β.
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