Higher order ${\mathcal{S}}^p$-differentiability: The unitary case

Abstract

Consider the set of unitary operators on a complex separable Hilbert space $\mathcal{H}$, denoted as $\mathcal{U}(\mathcal{H})$. Consider $1<p<\infty$. We establish that a function $f$ defined on the unit circle $\mathbb{T}$ is $n$ times continuously Fréchet ${\mathcal{S}}^p$-differentiable at every point in $\mathcal{U}(\mathcal{H})$ if and only if $f\in C^n(\mathbb{T})$. Take a function $U:\mathbb{R}\to \mathcal{U}(\mathcal{H})$ such that the function $t\in \mathbb{R}\mapsto U(t)-U(0)$ takes values in ${\mathcal{S}}^p$ and is $n$ times continuously ${\mathcal{S}}^p$-differentiable on $\mathbb{R}$. Consequently, for $f\in C^n(\mathbb{T})$, we prove that $f$ is $n$ times continuously Gâteaux ${\mathcal{S}}^p$-differentiable at $U(t)$. We provide explicit expressions for both types of derivatives of $f$ in terms of multiple operator integrals. In the domain of unitary operators, these results closely follow the $n$-th order successes for self-adjoint operators achieved by the second author, Le Merdy, Skripka, and Sukochev. Furthermore, as for application, we derive a formula and ${\mathcal{S}}^p$-estimates for operator Taylor remainders for a broader class of functions. Our results extend those of Peller, Potapov, Skripka, Sukochev, and Tomskova.