On positivity of spectral shift functions

We show that every spectral shift function of an even order η2k is nonnegative outside the convex hull of the spectrum of an initial operator cvhσ(H); every spectral shift function of an odd order η2k−1 is nonnegative (respectively, nonpositive) outside cvhσ(H) whenever a perturbation is nonnegative (respectively, nonpositive). We also derive several sufficient conditions for positivity of η2k and η2k−1 on the whole real line.