A noncommutative version of the bispectral problem. (Q1412819)

The second-order difference operator with matrix coefficients \(\mathcal{L}=E+B_j+A_jE^{-1}\), where \(Ef(j)=f(j+1)\), can be regarded as a doubly infinite tridiagonal matrix. The authors study the following matrix-valued bispectral problem. Problem: Find all nontrivial matrix-valued functions \(\Phi _j(z)\) depending on two variables \(j\) (discrete) and \(z\) (continuous), satisfying the following equations: \(\mathcal{L} \Phi _j(z)=z\Phi _j(z)\) \(\mathcal{B}(z,\partial _z)\Phi _j(z)^t=\Phi _j(z)^t\Lambda _j^t\) where \(\mathcal{B}(z,\partial _z)\) is a differential operator with matrix coefficients and \(\Lambda _j\) is some matrix. The derived necessary conditions for bispectrality and the presented examples show how much richer and complex the matrix valued situation is compared to the scalar one. The obtained results may be useful in the study of a noncommutative version of the nonlinear Toda lattice.