ABSTRACT. This paper resolves affirmatively Koplienko's conjecture of 1984 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 func-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 in 1953 and Koplienko in 1984, respectively, whereas for n > 2 the problem was unresolved. We show that η n,H,V exists, integrable, andfor 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.

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We construct higher order spectral shift functions, extending the perturbation theory results of M.G. Krein [M.G. Krein, On a trace formula in perturbation theory, Mat. Sb. 33 (1953) 597-626 (in Russian)] and L.S. Koplienko [L.S. Koplienko, Trace formula for perturbations of nonnuclear type, Sibirsk. Mat. Zh. 25 (1984) 62-71 (in Russian); translation in: Trace formula for nontrace-class perturbations, Siberian Math. J. 25 (1984) 735-743] on representations for the remainders of the first and second order Taylor-type approximations of operator functions. The higher order spectral shift functions represent the remainders of higher order Taylor-type approximations; they can be expressed recursively via the lower order (in particular, Krein's and Koplienko's) ones. We also obtain higher order spectral averaging formulas generalizing the Birman-Solomyak spectral averaging formula. The results are obtained in the semi-finite von Neumann algebra setting, with the perturbation taken in the Hilbert-Schmidt class of the algebra.

We prove perturbation results for traces on normed ideals in semifinite von Neumann algebra factors. This includes the case of Dixmier traces. In particular, we establish existence of spectral shift measures with initial operators being dissipative or bounded, and show that these measures can have singular components in the case of Dixmier traces. We also establish a linearization formula for a Dixmier trace applied to perturbed operator functions, a result that does not typically hold for normal traces.2000 Mathematics Subject Classification. Primary 47B10, secondary 47A55, 47L20.

Let A be a selfadjoint operator in a separable Hilbert space, K a selfadjoint Hilbert-Schmidt operator, and f ∈ C n (R). We establish that ϕ(t) = f (A + tK) − f (A) is n-times continuously differentiable on R in the Hilbert-Schmidt norm, provided either A is bounded or the derivatives f (i) , i = 1, . . . , n, are bounded. This substantially extends the results of [3] on higher order differentiability of ϕ in the Hilbert-Schmidt norm for f in a certain Wiener class. As an application of the second order S 2 -differentiability, we extend the Koplienko trace formula from the Besov class B 2 ∞1 (R) [20] to functions f for which the divided difference f [2] admits a certain Hilbert space factorization.