A ] 2 6 N ov 2 00 5 The determinant of AA ∗ − A ∗ A for a Leonard pair A , A ∗

Let K denote a field, and let V denote a vector space over K 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 = dimV −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∗.