Higher order Sᵖ-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 Sᵖ -differentiable at every point in U(H) if and only if f∈ Cⁿ(T) . Take a function U (H) such that the function t∈↦ U(t)-U(0) takes values in Sᵖ and is n times continuously Sᵖ -differentiable on . Consequently, for f∈ Cⁿ(T) , we prove that f is n times continuously Gâteaux Sᵖ -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 -estimates for operator Taylor remainders for a broader class of functions. Our results extend those of Peller, Potapov, Skripka, Sukochev, and Tomskova.