Spectral shift function of higher order

By D. Potapov, A. Skripka, Fedor Sukochev. 2009

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AbstractThis 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 ηn=ηn,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 Sn. Existence and summability of η1 and η2 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 ηn,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 cn depending only on n∈ℕ. Our results for ηn 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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e6056816792c92935ef619fce5fa713b98c89f23

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In Applied Mathematics

  1. V. Peller
  2. Richard C. Bradley
  3. Sandrine Grellier
  4. Fedor Sukochev
  5. Gregory Beylkin
  6. Lucas Monzón
  7. Patrick Gérard

In Mathematical Physics

  1. V. Peller

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