Simon B.
Spectral Averaging and the Krein Spectral Shift
(13K, AMSTeX)
ABSTRACT. We provide a new proof of a theorem of Birman and Solomyak that if $A(s) = A_0 +
sB$ with $B\geq 0$ trace class and $d\mu_s (\cdot) = \text{Tr}(B^{1/2} E_{A(s)}(\cdot)
B^{1/2})$, then $\int^1_0 [d\mu_s (\lambda)]\, ds = \xi(\lambda)\, d\lambda$ where $\xi$ is
the Krein spectral shift from $A(0)$ to $A(1)$. Our main point is that this is a simple
consequence of the formula: $\frac{d}{ds} \text{Tr}(f(A(s))=\text{Tr}(Bf'(A(s)))$.