Functions of unitaries with $\mathcal{S}^p$-perturbations for non continuously differentiable functions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929391929393152 |
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| author | Coine, Clément |
| author_facet | Coine, Clément |
| contents | 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 $1<p<\infty$ and let $\mathcal{S}^p(\mathcal{H})$ be the Schatten class of order $p$ on $\mathcal{H}$. If $\tilde{U}:t\in\mathbb{R} \mapsto U(t)-U(0)$ is $n$-times $\mathcal{S}^p$-differentiable on $\mathbb{R}$, we show that the operator valued function $φ: t\in \mathbb{R} \mapsto f(U(t)) - f(U(0)) \in \mathcal{S}^p(\mathcal{H})$ 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 $φ$ are given in terms of multiple operator integrals and a formula and $\mathcal{S}^p$-estimates for the Taylor remainders of $φ$ are provided. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_13333 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Functions of unitaries with $\mathcal{S}^p$-perturbations for non continuously differentiable functions Coine, Clément Functional Analysis 47B49, 47B10, 46L52, 47A55 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 $1<p<\infty$ and let $\mathcal{S}^p(\mathcal{H})$ be the Schatten class of order $p$ on $\mathcal{H}$. If $\tilde{U}:t\in\mathbb{R} \mapsto U(t)-U(0)$ is $n$-times $\mathcal{S}^p$-differentiable on $\mathbb{R}$, we show that the operator valued function $φ: t\in \mathbb{R} \mapsto f(U(t)) - f(U(0)) \in \mathcal{S}^p(\mathcal{H})$ 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 $φ$ are given in terms of multiple operator integrals and a formula and $\mathcal{S}^p$-estimates for the Taylor remainders of $φ$ are provided. |
| title | Functions of unitaries with $\mathcal{S}^p$-perturbations for non continuously differentiable functions |
| topic | Functional Analysis 47B49, 47B10, 46L52, 47A55 |
| url | https://arxiv.org/abs/2406.13333 |