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=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∗.