Multiple operator integrals and higher operator derivatives

In this paper we consider the problem of the existence of higher derivatives of the function t↦ϕ(A+tK), where ϕ is a function on the real line, A is a self-adjoint operator, and K is a bounded self-adjoint operator. We improve earlier results by Sten’kin. In order to do this, we give a new approach to multiple operator integrals. This approach improves the earlier approach given by Sten’kin. We also consider a similar problem for unitary operators.