Functions of unitaries with $\mathcal{S}^p$-perturbations for non-continuously differentiable functions
Streszczenie
Consider a function $f : \mathbb {T} \to \mathbb {C}$, $n$-times differentiable on $\mathbb {T}$ and such that its $n$th derivative $f^{(n)}$ is bounded but not necessarily continuous. Let $U : \mathbb {R} \to \mathcal {U}(\mathcal {H})$ be a function taking values in the set of unitary operators on some separable Hilbert space $\mathcal {H}$. Let $1 \lt p \lt \infty $ and let $\mathcal {S}^p(\mathcal {H})$ be the Schatten class of order $p$ on $\mathcal {H}$. If $\tilde {U}: \mathbb {R}\ni t \mapsto U(t)-U(0)$ is $n$-times $\mathcal {S}^p$-differentiable on $\mathbb {R}$, we show that the operator-valued function $\varphi : \mathbb {R}\ni t \mapsto f(U(t)) - f(U(0)) \in \mathcal {S}^p(\mathcal {H})$ is $n$-times differentiable on $\mathbb {R}$ as well. This theorem is optimal and extends several results related to the differentiability of functions of unitaries. The derivatives of $\varphi $ are given in terms of multiple operator integrals, and a formula and $\mathcal {S}^p$-estimates for the Taylor remainders of $\varphi $ are provided.
