A canonical factorization for meromorphic Herglotz functions on the unit disk and sum rules for Jacobi matrices

We prove a general canonical factorization for meromorphic Herglotz functions on the unit disk whose notable elements are that there is no restriction (other than interlacing) on the zeros and poles for their Blaschke product to converge and there is no singular inner function. We use this result to provide a significant simplification in the proof of Killip-Simon (Ann. Math. 158 (2003) 253) of their result characterizing the spectral measures of Jacobi matrices, J, with J−J₀ Hilbert-Schmidt. We prove a nonlocal version of Case and step-by-step sum rules.