We discuss the proof of and systematic application of Case's sum rules for Jacobi matrices. Of special interest is a linear combination of two of his sum rules which has strictly positive terms. Among our results are a complete classification of the spectral measures of all Jacobi matrices J for which J -J 0 is Hilbert-Schmidt, and a proof of Nevai's conjecture that the Szeg condition holds if J -J 0 is trace class.
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Killip et al. (2003) studied this question.
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