Eigenfunction Expansion and the Decomposition of Jacobi Operators on $\mathbb{Z}$

Let $J$ be a Jacobi operator on ${\mathrm{\ell }}^2\left(\mathbb{Z}\right)$. We prove an eigenfunction expansion theorem for the singular part of $J$ using subordinate solutions to the eigenvalue equation. We exploit this theorem in order to show that $J$ can be decomposed as a direct integral of half-line operators.