Spectral shift function of higher order

Abstract

This paper resolves affirmatively Koplienko’s (Sib. Mat. Zh. 25:62–71, 1984) conjecture on existence of higher order spectral shift measures. Moreover, the paper establishes absolute continuity of these measures and, thus, existence of the higher order spectral shift functions. A spectral shift function of order n∈ℕ is the function η _n =η _n,H,V such that

$$ \operatorname {Tr}\Biggl( f(H + V)-\sum_{k = 0}^{n-1} \frac{1}{k!}\, \frac{d^k}{dt^k} \bigl[ f(H + tV) \bigr] \bigg|_{t = 0} \Biggr) = \int_\mathbb{R}f^{(n)} (t)\, \eta_n (t)\, dt, $$

for every sufficiently smooth function f, where H is a self-adjoint operator defined in a separable Hilbert space ℍ and V is a self-adjoint operator in the n-th Schatten-von Neumann ideal S ⁿ. Existence and summability of η ₁ and η ₂ were established by Krein (Mat. Sb. 33:597–626, 1953) and Koplienko (Sib. Mat. Zh. 25:62–71, 1984), respectively, whereas for n>2 the problem was unresolved. We show that η _n,H,V exists, integrable, and

$$\Vert \eta_n \Vert _{L^1(\mathbb{R})} \leq c_n \Vert V \Vert _{S^n}^n, $$

for some constant c _n depending only on n∈ℕ. Our results for η _n rely on estimates for multiple operator integrals obtained in this paper. Our method also applies to the general semi-finite von Neumann algebra setting of the perturbation theory.