HIGHER ORDER DIFFERENTIABILITY OF OPERATOR FUNCTIONS IN SCHATTEN NORMS

We establish the following results on higher order ${\mathcal{S}}^{p}$ -differentiability, $1&lt;p&lt;\infty$ , 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 $f\in C^{n}(\mathbb{R})$ ; (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 $f\in B_{\infty 1}^{n}(\mathbb{R})\cap B_{\infty 1}^{1}(\mathbb{R})$ , then $f$ is $n$ times continuously Fréchet ${\mathcal{S}}^{q}$ -differentiable, $1\leqslant q&lt;\infty$ , 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 $f\in C^{n}(\mathbb{R})$ with $f^{\prime },\ldots ,f^{(n)}\in C_{b}(\mathbb{R})$ , and Gâteaux <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S14

Hali tarjima qilinmagan