Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the split decomposition

Abstract

Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. We consider an ordered pair of linear transformations $A:V\to V$ and $A^*:V\to V$ that satisfy conditions (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 diagonal and the matrix representing $A^*$ is irreducible tridiagonal. We call such a pair a {\it Leonard pair} on $V$. Let $A,A^*$ denote a Leonard pair on $V$. There exists a decomposition of $V$ into a direct sum of 1-dimensional subspaces, with respect to which $A$ is lower bidiagonal and $A^*$ is upper bidiagonal. This is known as the {\it split decomposition}. We use the split decomposition to obtain several characterizations of Leonard pairs.