Linear transformations that are tridiagonal with respect to both eigenbases of a Leonard pair

Abstract

Let V denote a vector space with finite positive dimension. We consider a pair of linear transformations A : V → V and A* : V → V that satisfy (i) and (ii) below:(i)There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A* is diagonal.(ii)There exists a basis for V with respect to which the matrix representing A* is irreducible tridiagonal and the matrix representing A is diagonal.We call such a pair a Leonard pair on V. Let X denote the set of linear transformations X : V → V such that the matrix representing X with respect to the basis (i) is tridiagonal and the matrix representing X with respect to the basis (ii) is tridiagonal. We show that X is spanned byI, A, A*, AA*, A* Aand these elements form a basis for X provided the dimension of V is at least 3. © 2006 Elsevier Inc. All rights reserved.