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 $\phi :t\in \mathbb{R}\mapsto f(U(t))-f(U(0))\in {\mathcal{S}}^p(\mathcal{H})$ 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 $\phi$ are given in terms of multiple operator integrals and a formula and ${\mathcal{S}}^p$-estimates for the Taylor remainders of $\phi$ are provided.

31 pages