The determinant of $AA^*-A^*A$ for a Leonard pair $A,A^*$

Abstract

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 $A:V \to V$ and $A^*: V \to V$ 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 {\em Leonard pair} on $V$. In this paper we investigate the commutator $AA^*-A^*A$. Our results are as follows. First assume the dimension of $V$ is even. We show $AA^*-A^*A$ is invertible and display several attractive formulae for the determinant. Next assume the dimension of $V$ is odd. 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 $A^*$.