A Cayley–Hamiltonian theorem for periodic finite band matrices

Abstract

Let $K$ be a doubly infinite, self-adjoint matrix which is finite band (i.e. $K_{jk} = 0$ if $|j-k| > m$) and periodic ($KS^n = S^nK$ for some $n$ where $(Su)_j = u_{j+1}$) and non-degenerate (i.e. $K_{j j+m} \ne 0$ 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 $p(x,y) = y^2 - y \Delta(x) + 1$ where $\Delta$ is the discriminant. I hope Pavel Exner will enjoy this birthday bouquet.