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 ($K{S}^n=S^{n}K$ for some $n$ where $(Su{)}_j=u_{j+1}$) and non-degenerate (i.e. $K_{jj+m}\ne 0$ 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.