Higher derivatives of operator functions in ideals of von Neumann algebras

TL;DR

This paper develops a framework for higher derivatives of operator functions within ideals of von Neumann algebras, introducing new function spaces and proving differentiability formulas in this context.

Contribution

It defines integral symmetrically normed ideals and introduces the space OC^{[k]}(R), establishing differentiability of operator functions in these ideals with explicit formulas.

Findings

Established that certain function spaces ensure differentiability of operator functions.

Proved that multiple operator integrals can express derivatives.

Identified classes of ideals where these results hold, including Schatten p-ideals and compact operators.

Abstract

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.