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

Consider the set of unitary operators on a complex separable Hilbert space $\text{hilh}$, denoted as $\mathcal{U}(\text{hilh})$. Consider $1<p<\mathrm{\infty }$. We establish that a function $f$ defined on the unit circle $\text{cir}$ is $n$ times continuously Fréchet ${\text{Sp}}^p$-differentiable at every point in $\mathcal{U}(\text{hilh})$ if and only if $f\in C^n(\text{cir})$. Take a function $U:\mathbb{R}\to \mathcal{U}(\text{hilh})$ such that the function $t\in \mathbb{R}\mapsto U(t)-U(0)$ takes values in ${\text{Sp}}^p$ and is $n$ times continuously ${\text{Sp}}^p$-differentiable on $\mathbb{R}$. Consequently, for $f\in C^n(\text{cir})$, 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 ${\text{Sp}}^p$-estimates for operator Taylor remainders for a broader class of functions. Our results extend those of Peller, Potapov, Skripka, Sukochev and Tomskova.