Higher derivatives of operator functions in ideals of von Neumann algebras

DOI10.1016/J.JMAA.2022.126705arXiv2107.03693MaRDI QIDQ6372296

Evangelos A. Nikitopoulos

Publication date: 8 July 2021

Abstract: Let $mathcalM$ be a von Neumann algebra and $a$ be a self-adjoint operator affiliated with $mathcalM$. We define the notion of an "integral symmetrically normed ideal" of $mathcalM$ and introduce a space $O{C}^{[k]}(mathbbR)subseteq{C}^k(mathbbR)$ of functions $mathbbRomathbbC$ such that the following result holds: for any integral symmetrically normed ideal $mathcalI$ of $mathcalM$ and any $finO{C}^{[k]}(mathbbR)$, the operator function $mathcal{I}_{mathrmsa}ibmapstof(a+b)-f(a)inmathcalI$ 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 $findot{B}_1^{1,infty}(mathbbR)capdot{B}_1^{k,infty}(mathbbR)$ is such that $f^{\prime }$ is bounded, then $finO{C}^{[k]}(mathbbR)$. Finally, we prove that all of the following ideals are integral symmetrically normed: $mathcalM$ itself, separable symmetrically normed ideals, Schatten $p$-ideals, the ideal of compact operators, and -- when $mathcalM$ is semifinite -- ideals induced by fully symmetric spaces of measurable operators.

Mathematics Subject Classification ID

General theory of von Neumann algebras (46L10) Functional calculus for linear operators (47A60) Noncommutative function spaces (46L52) Operator ideals (47L20)

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