  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 Szego
condition holds if J-J_0 is trace class.
