LAUSR

Abstract We show that every spectral shift function of an even order η 2 k is nonnegative outside the convex hull of the spectrum of an initial operator cvh σ ( H ) ; every spectral shift function of an odd order η 2 k − 1 is nonnegative (respectively, nonpositive) outside cvh σ ( H ) whenever a perturbation is nonnegative (respectively, nonpositive). We also derive several sufficient conditions for positivity of η 2 k and η 2 k − 1 on the whole real line.