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- W3100655207 abstract "We establish the following results on higher order ${mathcal{S}}^{p}$ -differentiability, $1<p<infty$ , of the operator function arising from a continuous scalar function $f$ and self-adjoint operators defined on a fixed separable Hilbert space: (i) $f$ is $n$ times continuously Fréchet ${mathcal{S}}^{p}$ -differentiable at every bounded self-adjoint operator if and only if $fin C^{n}(mathbb{R})$ ; (ii) if $f^{prime },ldots ,f^{(n-1)}in C_{b}(mathbb{R})$ and $f^{(n)}in C_{0}(mathbb{R})$ , then $f$ is $n$ times continuously Fréchet ${mathcal{S}}^{p}$ -differentiable at every self-adjoint operator; (iii) if $f^{prime },ldots ,f^{(n)}in C_{b}(mathbb{R})$ , then $f$ is $n-1$ times continuously Fréchet ${mathcal{S}}^{p}$ -differentiable and $n$ times Gâteaux ${mathcal{S}}^{p}$ -differentiable at every self-adjoint operator. We also prove that if $fin B_{infty 1}^{n}(mathbb{R})cap B_{infty 1}^{1}(mathbb{R})$ , then $f$ is $n$ times continuously Fréchet ${mathcal{S}}^{q}$ -differentiable, $1leqslant q<infty$ , at every self-adjoint operator. These results generalize and extend analogous results of Kissin et al. ( Proc. Lond. Math. Soc. (3) 108 (3) (2014), 327–349) to arbitrary $n$ and unbounded operators as well as substantially extend the results of Azamov et al. ( Canad. J. Math. 61 (2) (2009), 241–263); Coine et al. ( J. Funct. Anal. ; doi: 10.1016/j.jfa.2018.09.005 ); Peller ( J. Funct. Anal. 233 (2) (2006), 515–544) on higher order ${mathcal{S}}^{p}$ -differentiability of $f$ in a certain Wiener class, Gâteaux ${mathcal{S}}^{2}$ -differentiability of $fin C^{n}(mathbb{R})$ with $f^{prime },ldots ,f^{(n)}in C_{b}(mathbb{R})$ , and Gâteaux ${mathcal{S}}^{q}$ -differentiability of $f$ in the intersection of the Besov classes $B_{infty 1}^{n}(mathbb{R})cap B_{infty 1}^{1}(mathbb{R})$ . As an application, we extend ${mathcal{S}}^{p}$ -estimates for operator Taylor remainders to a broad set of symbols. Finally, we establish explicit formulas for Fréchet differentials and Gâteaux derivatives." @default.
- W3100655207 created "2020-11-23" @default.
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- W3100655207 date "2019-02-13" @default.
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- W3100655207 title "HIGHER ORDER DIFFERENTIABILITY OF OPERATOR FUNCTIONS IN SCHATTEN NORMS" @default.
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- W3100655207 doi "https://doi.org/10.1017/s1474748019000033" @default.
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