Higher derivatives of operator functions in ideals of von Neumann algebras

Let M be a von Neumann algebra and a be a self-adjoint operator affiliated with M. We define the notion of an “integral symmetrically normed ideal” of M and introduce a space OC[k](R)⊆Ck(R) of functions R→C such that the following holds: for any integral symmetrically normed ideal I of M and any f∈O...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications Vol. 519; no. 1; p. 126705
Main Author: Nikitopoulos, Evangelos A.
Format: Journal Article
Language:English
Published: Elsevier Inc 01.03.2023
Subjects:
Fréchet derivative
Functional calculus
Measurable operator
Multiple operator integral
Noncommutative Banach function space
Symmetrically normed ideal
Noncommutative Banach function space
Multiple operator integral
Measurable operator
Fréchet derivative
Symmetrically normed ideal
Functional calculus
ISSN:0022-247X, 1096-0813
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Summary:Let M be a von Neumann algebra and a be a self-adjoint operator affiliated with M. We define the notion of an “integral symmetrically normed ideal” of M and introduce a space OC[k](R)⊆Ck(R) of functions R→C such that the following holds: for any integral symmetrically normed ideal I of M and any f∈OC[k](R), the operator function Isa∋b↦f(a+b)−f(a)∈I is k-times continuously Fréchet differentiable, and the formula for its derivatives may be written in terms of multiple operator integrals. Moreover, we prove that if f∈B˙11,∞(R)∩B˙1k,∞(R) and f′ is bounded, then f∈OC[k](R). Finally, we prove that all of the following ideals are integral symmetrically normed: M itself, separable symmetrically normed ideals, Schatten p-ideals, the ideal of compact operators, and – when M is semifinite – ideals induced by fully symmetric spaces of measurable operators.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2022.126705