The switching element for a Leonard pair

Let V denote a vector space with finite positive dimension. We consider a pair of linear transformations A:V→V and A∗:V→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 (resp. ) denote a basis for V referred to in (i) (resp. (ii)). We show that there exists a unique linear transformation S:V→V that sends v0 to a scalar multiple of vd, fixes w0, and sends wi to a scalar multiple of wi for 1⩽i⩽d. We call S the switching element. We describe S from many points of view.