  We use the classical results of Baxter and Gollinski-Ibragimov to prove a new
spectral equivalence for Jacobi matrices on $l^2(\N)$. In particular, we
consider the class of Jacobi matrices with conditionally summable parameter
sequences and find necessary and sufficient conditions on the spectral measure
such that $\sum_{k=n}^\infty b_k$ and $\sum_{k=n}^\infty (a_k^2 - 1)$ lie in
$l^2_1 \cap l^1$ or $l^1_s$ for $s \geq 1$.
