A Cayley-Hamiltonian Theorem for Periodic Finite Band Matrices

Let K be a doubly infinite, self-adjoint matrix which is finite band (i.e. K_(jk) = 0 if |j – k| > m) and periodic (K S^n = S^n K for some n where (Su)_j = u_(j+1)) and non-degenerate (i.e. K_(jj+m) ≠ = 0 for all j). Then, there is a polynomial, p(x, y), in two variables with p(K, S^n) = 0. This generalizes the tridiagonal case where p(x, y) = y^2 - yΔ(x) + 1 where Δ is the discriminant. I hope Pavel Exner will enjoy this birthday bouquet.