Higher order $\mathcal{S}^{p}$-differentiability: The unitary case
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| Format: | Preprint |
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2024
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| _version_ | 1866910651836792832 |
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| author | Chattopadhyay, Arup Coine, Clément Giri, Saikat Pradhan, Chandan |
| author_facet | Chattopadhyay, Arup Coine, Clément Giri, Saikat Pradhan, Chandan |
| contents | 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 Fréchet $\Sp^p$-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 Gâteaux $\mathcal{S}^p$-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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_08253 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Higher order $\mathcal{S}^{p}$-differentiability: The unitary case Chattopadhyay, Arup Coine, Clément Giri, Saikat Pradhan, Chandan Functional Analysis 47B49, 47B10, 46L52, 47A55 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 Fréchet $\Sp^p$-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 Gâteaux $\mathcal{S}^p$-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. |
| title | Higher order $\mathcal{S}^{p}$-differentiability: The unitary case |
| topic | Functional Analysis 47B49, 47B10, 46L52, 47A55 |
| url | https://arxiv.org/abs/2404.08253 |