Characterizing when divided differences are valid symbols for multiple operator integrals

Determine general, practically checkable conditions on a scalar function f that guarantee its n-th divided difference Dif_n f belongs to the symbol space S((E_0,...,E_n),(T_1,...,T_n);ψ) associated with multiple operator integrals J(·, (E_0,...,E_n), (T_1,...,T_n)) on a separable Hilbert space H. Concretely, for arbitrary tuples of spectral measures (E_0,...,E_n) and operator vectors (T_1,...,T_n) as defined in Definition 2.1, identify sufficient and ideally necessary regularity or integrability assumptions on f ensuring Dif_n f ∈ S((E_0,...,E_n),(T_1,...,T_n);ψ) so that the integral is well-defined, beyond the known sufficient condition that f^(n) has an integrable Fourier transform.

Background

Multiple operator integrals (MOI) provide a functional calculus for several spectral measures and play a central role in non-commutative Taylor expansions and the construction of higher order spectral shift functions. The MOI J(φ, E, T) requires its symbol φ to belong to a symbol space S(E,T;ψ), characterized by an integral factorization against measurable kernels (Definition 2.1).

In applications, φ is typically the divided difference Dif_n f of a scalar function f. While the paper supplies a widely used sufficient condition—namely, that the n-th derivative f^n has an integrable Fourier transform—this does not settle the broader question of finding general, accessible criteria on f that ensure Dif_n f lies in the symbol space for MOI with arbitrary spectral data.

The author notes that, despite considerable development of MOI theory, a general characterization remains out of reach, and points to Peller’s survey for further context. A resolution would streamline assumptions in perturbation theory and broaden the applicability of MOI beyond currently known function classes.