Linear transformations that are tridiagonal with respect to both eigenbases of a Leonard pair

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) and (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 {\it Leonard pair} on $V$. Let $\cal X$ denote the set of linear transformations $X:V \to V$ such that the matrix representing $X$ with respect to the basis (i) is tridiagonal and the matrix representing $X$ with respect to the basis (ii) is tridiagonal. We show that $\cal X$ is spanned by $I$, $A$, $A^*$, $AA^*$, $A^*A$, and these elements form a basis for $\cal X$ provided the dimension of $V$ is at least 3.