A characterization of Leonard pairs using the notion of a tail

Abstract

Let $V$ denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations $A:V\to V$ and $A^{\ast }:V\to V$ 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 $A^{\ast }$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^{\ast }$ is irreducible tridiagonal and the matrix representing $A$ is diagonal. We call such a pair a Leonard pair on $V$. In this paper, we characterize the Leonard pairs using the notion of a tail. This notion is borrowed from algebraic graph theory.

12 pages

References in corpus (1)

• Two linear transformations each tri-diagonal with respect to an eigenbasis of the other; the TD-D canonical form and the LB-UB canonical form