Higher derivatives of operator functions in ideals of von Neumann algebras

• Published in: arXiv.org 2023-12
• Main Author: Nikitopoulos, Evangelos A
• Format: Article
• Language: English
• Subjects: Derivatives; Integrals; Mathematical analysis; Operators (mathematics)
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2107.03693

Let \(\mathscr{M}\) be a von Neumann algebra and \(a\) be a self-adjoint operator affiliated with \(\mathscr{M}\). We define the notion of an "integral symmetrically normed ideal" of \(\mathscr{M}\) and introduce a space \(OC^{[k]}(\mathbb{R}) \subseteq C^k(\mathbb{R})\) of functions \(\mathbb{R} \to \mathbb{C}\) such that the following result holds: for any integral symmetrically normed ideal \(\mathscr{I}\) of \(\mathscr{M}\) and any \(f \in OC^{[k]}(\mathbb{R})\), the operator function \(\mathscr{I}_{\mathrm{sa}} \ni b \mapsto f(a+b)-f(a) \in \mathscr{I}\) is \(k\)-times continuously Fr\'{e}chet differentiable, and the formula for its derivatives may be written in terms of multiple operator integrals. Moreover, we prove that if \(f \in \dot{B}_1^{1,\infty}(\mathbb{R}) \cap \dot{B}_1^{k,\infty}(\mathbb{R})\) and \(f'\) is bounded, then \(f \in OC^{[k]}(\mathbb{R})\). Finally, we prove that all of the following ideals are integral symmetrically normed: \(\mathscr{M}\) itself, separable symmetrically normed ideals, Schatten \(p\)-ideals, the ideal of compact operators, and -- when \(\mathscr{M}\) is semifinite -- ideals induced by fully symmetric spaces of measurable operators.