 99339 F. Gesztesy, K. A. Makarov, and A. K. Motovilov
 Monotonicity and Concavity Properties of The Spectral Shift Function
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Sep 14, 99

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Abstract. Let H_0 and V(s) be selfadjoint, V, V' continuously differentiable in trace norm with V''(s)\geq 0 for s\in (s_1,s_2), and denote by
{E_{H(s)}(\lambda)}_{\lambda\in\bbR} the family of spectral projections of H(s)=H_0+V(s). Then we prove for given \mu\in\bbR, that s\longmapsto
\tr (V'(s)E_{H(s)}((\infty, \mu))) is a nonincreasing function with respect to s, extending a result of Birman and Solomyak. Moreover, denoting by
\zeta (\mu,s)=\int_{\infty}^\mu d\lambda \xi(\lambda,H_0,H(s)) the integrated spectral shift function for the pair (H_0,H(s)), we prove concavity of
\zeta (\mu,s) with respect to s, extending previous results by Geisler, Kostrykin, and Schrader. Our proofs employ operatorvalued Herglotz functions and establish the latter as an effective tool in this context.
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