The determinant of AA*–A*A for a Leonard pair A, A*

Let denote a field, and let V denote a vector space over with finite positive dimension. We consider a pair of linear transformations A: V → V and A*: V → V that satisfy (i), (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. In this paper we investigate the commutator AA* − A*A. Our results are as follows. Abbreviate d=dim V − 1 and first assume d is odd. We show AA* − A*A is invertible and display several attractive formulae for the determinant. Next assume d is even. We show that the null space of AA* − A*A has dimension 1. We display a nonzero vector in this null space. We express this vector as a sum of eigenvectors for A and as a sum of eigenvectors for A*.