Perturbation theory and higher order $\mathcal{S}^{\!p}$-differentiability of operator functions

We establish, for $1<p<\\infty$, higher order $\\mathcal{S}^{p}$-differentiability results of the function $\\varphi \\colon t\\in \\mathbb{R} \\mapsto f(A+tK) - f(A)$ selfadjoint operators $A$ and $K$ on a separable Hilbert space $\\mathcal{H}$ with element Schatten class $\\mathcal{S}^{p}(\\mathcal{H})$ $f$ $n$-times differentiable $\\mathbb{R}$. prove that if either $f^{(n)}$ are bounded, or $f^{(i)}$, $1\\leq i\\leq n$, $\\varphi$ is $\\mathbb{R}$ in $\\mathcal{S}^{p}$-norm bounded $n$th derivative. If $f\\in C^n(\\mathbb{R})$ $f^{(n)}$, we continuously give explicit formulas derivatives $\\varphi$, terms multiple operator integrals. As application, establish formula $\\mathcal{S}^{p}$-estimates Taylor remainders more extensive functions. These analogue by Kissin–Potapov–Shulman–Sukochev. They also extend Le Merdy–Skripka from functions to $f$.