The determinant of \(AA^{*} - A^{*}A\) for a Leonard pair \(A,A^{*}\) (Q2496658)

Let \(V\) denote a finite-dimensional vector space over any field, dim\(\,V=:d-1\). A Leonard pair \((A, A^*)\) is a pair of linear transformations \(A: V\to V\) and \(A^*: V\to V\) with the following properties: (i) There exists a basis of \(V\) such that the matrices of \(A\) and \(A^*\) with respect to this basis are irreducible tridiagonal and diagonal, respectively. (ii) There exists a basis of \(V\) such that the matrices of \(A^*\) and \(A\) with respect to this basis are irreducible tridiagonal and diagonal, respectively. (Here `irreducible tridiagonal' means that all entries in upper and lower parallel to the main diagonal are non-zero.) In this article, the commutator \(AA^*-A^*A\) is investigated. The main result is as follows. If \(d\) is odd then \(AA^*-A^*A\) is invertible. If \(d\) is even then the kernel (null space) of \(AA^*-A^*A\) has dimension \(1\). Furthermore, the determinant of \(AA^*-A^*A\) is given for \(d\) odd, and a basis of the kernel of \(AA^*-A^*A\) for \(d\) even.