Functions of unitaries with $\mathcal{S}^p$-perturbations for non continuously differentiable functions

Abstract

Consider a function $f : \mathbb{T} \to \mathbb{C}$, $n$-times differentiable on $\mathbb{T}$ and such that its $n$th derivative $f^{(n)}$ is bounded but not necessarily continuous. Let $U : \mathbb{R} \to \mathcal{U}(\mathcal{H})$ be a function taking values in the set of unitary operators on some separable Hilbert space $\mathcal{H}$. Let $1<p<\infty$ and let $\mathcal{S}^p(\mathcal{H})$ be the Schatten class of order $p$ on $\mathcal{H}$. If $\tilde{U}:t\in\mathbb{R} \mapsto U(t)-U(0)$ is $n$-times $\mathcal{S}^p$-differentiable on $\mathbb{R}$, we show that the operator valued function $\varphi : t\in \mathbb{R} \mapsto f(U(t)) - f(U(0)) \in \mathcal{S}^p(\mathcal{H})$ is $n$-times differentiable on $\mathbb{R}$ as well. This theorem is optimal and extends several results related to the differentiability of functions of unitaries. The derivatives of $\varphi$ are given in terms of multiple operator integrals and a formula and $\mathcal{S}^p$-estimates for the Taylor remainders of $\varphi$ are provided.