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 $\phi :t\in \mathbb{R}\mapsto f(A+tK)-f(A)$ 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 $f^{(i)}$, $1\le i\le n$, are bounded, $\phi$ 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 $\phi$ is $n$-times continuously differentiable on $\mathbb{R}$. We give explicit formulas for the derivatives of $\phi$, 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 $f$.