Affine transformations of a Leonard pair

Abstract

Let K denote a field and let V denote a vector space over K withfinite positive
dimension. An ordered pair is considered of linear transformations A : V → V and A^∗ : V → V that
satisfy (i) and (ii) below:

(i) There exists a basis for V withrespect to whichthe matrix representing A is irreducible
tridiagonal and the matrix representing A^∗ is diagonal.

(ii) There exists a basis for V withrespect to whichthe matrix representing A^∗ is irreducible
tridiagonal and the matrix representing A is diagonal.

Sucha pair is called a Leonard pair on V . Let ξ, ζ, ξ^∗, ζ^∗ denote scalars in K with ξ, ξ^∗ nonzero, and
note that ξA + ζI, ξ^∗A∗ + ζ^∗I is a Leonard pair on V . Necessary and sufficient conditions are given
for this Leonard pair to be isomorphic to A, A^∗. Also given are necessary and sufficient conditions
for this Leonard pair to be isomorphic to the Leonard pair A^∗, A.