Estimates and trace formulas for unitary and resolvent comparable perturbations

We prove estimates for Schatten norms of multiple operator integrals with symbols arising from Fourier series and, as a consequence, derive higher order trace formulas for functions given by absolutely convergent Fourier series of unitary operators and for rational functions bounded on the real line of resolvent comparable self-adjoint operators. These results substantially extend the main results of [12,14,13], where the estimates were proved for multiple operator integrals with symbols arising from polynomials, the trace formulas in the unitary case for analytic functions on the unit disc, and the trace formulas in the resolvent comparable self-adjoint case for rational functions bounded and analytic in the lower half-plane. Our results rely crucially on an algebraic approach to symbols of multiple operator integrals developed in this paper to replace the prior restrictive analytic approach of [12].