Higher order S^p-differentiability: The unitary case

Consider the set of unitary operators on a complex separable Hilbert space H, denoted as U (H). Consider 1<p<. We establish that a function f defined on the unit circle T is n times continuously Fréchet S^p -differentiable at every point in U (H) if and only if f C^n (T). Take a function U (H) such that the function t U (t) -U (0) takes values in S^p and is n times continuously S^p -differentiable on. Consequently, for f C^n (T), we prove that f is n times continuously Gâteaux 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 S^p -estimates for operator Taylor remainders for a broader class of functions. Our results extend those of Peller, Potapov, Skripka, Sukochev, and Tomskova.