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

Consider the set of unitary operators on a complex separable Hilbert space $\hilh$, denoted as $\mathcal{U}(\hilh)$. Consider $1<p<\infty$. We establish that a function $f$ defined on the unit circle $\cir$ is $n$ times continuously Fr\'echet $\Sp^p$-differentiable at every point in $\mathcal{U}(\hilh)$ if and only if $f\in C^n(\cir)$. Take a function $U :\R\rightarrow\mathcal{U}(\hilh)$ such that the function $t\in\R\mapsto U(t)-U(0)$ takes values in $\Sp^{p}$ and is $n$ times continuously $\Sp^{p}$-differentiable on $\R$. Consequently, for $f\in C^n(\cir)$, we prove that $f$ is $n$ times continuously G\^ateaux $\mathcal{S}^p$-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 $\Sp^{p}$-estimates for operator Taylor remainders for a broader class of functions. Our results extend those of Peller, Potapov, Skripka, Sukochev and Tomskova.