Monotonicity and concavity properties of the spectral shift function (Q2702388)

Let \(H_0\) and \(V(s)\) be selfadjoint operators in a complex separable Hilbert space, \(V, V'\) continuously differentiable in trace norm with \(V''(s)\geq 0\) for \(s\in (s_1,s_2)\), and denote by \(\{ E_{H(s)}(\lambda)\}_{\lambda\in{\mathbb R}}\) the family of spectral projections of \(H(s)=H_0+V(s)\). It is proved that for given \(\mu\in {\mathbb R}\) function \(s\to tr (V'(s)E_{H(s)}((-\infty, \mu)))\) is nonincreasing with respect to \(s\). It is extension of a \textit{M. Š. Birman} and \textit{M. Z. Solomyak} result [see J. Soc. Math. 3, 408--419 (1975; Zbl 0336.47017)] proved for the case of \(V(s)=sV\). NEWLINENEWLINENEWLINEDenote by \(\xi(\lambda,H,H_0)\) the spectral shift function for the pair \((H,H_0)\). Let NEWLINE\[NEWLINE\zeta (\mu,s)=\int_{-\infty}^\mu d\lambda \xi(\lambda,H_0,H(s))NEWLINE\]NEWLINE be the integrated spectral shift function for the pair \((H_0,H(s))\). Concavity of \(\zeta (\mu,s)\) with respect to \(s\) is also proved. This result is extension of previous results by \textit{R. Geisler, V. Kostrykin}, and \textit{R. Schrader} [see Rev. Math. Phys. 7, No.~2, 161--181 (1995; Zbl 0836.47049)]. Proofs employ operator-valued Herglotz functions.NEWLINENEWLINEFor the entire collection see [Zbl 0953.00049].