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

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 → V and A * : V → V that satisfy both conditions below:(i) exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A * is diagonal. exists a basis for V with respect to which the matrix representing A * is irreducible tridiagonal and the matrix representing A is diagonal. l such a pair a Leonard pair on V. Referring to the above Leonard pair, it is known there exists a decomposition of V into a direct sum of one-dimensional subspaces, on which A acts in a lower bidiagonal fashion and A * acts in an upper bidiagonal fashion. This is called the split decomposition. In this paper, we give two characterizations of a Leonard pair that involve the split decomposition.