Affine transformations 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 an ordered 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 Leonard pair on $V$. Let $x$, $c$, $x^*$, $c^*$ denote scalars in $K$ with $x$, $x^*$ nonzero, and note that $xA+cI$, $x^*A^* + c^*I$ is a Leonard pair on $V$. We give necessary and sufficient conditions for this Leonard pair to be isomorphic to the Leonard pair $A$, $A^*$. We also give necessary and sufficient conditions for this Leonard pair to be isomorphic to the Leonard pair $A^*$, $A$.