Spectral averaging and the Krein spectral shift. (English) Zbl 0892.47021
Summary: 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 \(\frac{d}{ds} \text{Tr}(f(A(s))=\text{Tr}(Bf'(A(s)))\).
MSC:
| 47B10 | Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.) |
| 47A60 | Functional calculus for linear operators |
References:
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