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

Abstract

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