Matrix Bispectrality of Full Rank One Algebras

Abstract

We study algebraic properties of full rank 1 algebras in a general framework and derive a method to verify if one such matrix polynomial sub-algebra is bispectral. We give two examples illustrating the method. In the first one, we consider the eigenvalue to be scalar-valued, whereas, in the second one, we assume it to be matrix-valued. In the former example, we put forth a Pierce decomposition of that algebra.