A noncommutative version of the bispectral problem

We consider a matrix valued version of the bispectral problem involving a block tridiagonal doubly infinite matrix and a first-order differential operator with matrix coefficients. We give a set of necessary conditions that the coefficients need to satisfy and solve these equations under a variety of conditions. The situations discussed here should make it plain that while the corresponding problem in the scalar case is relatively trivial and devoid of any interest, the noncommutative version of the problem is much richer and subtle. The results here should be useful, for instance, in the study of a noncommutative version of the nonlinear Toda lattice.