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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Veröffentlicht in:Journal of mathematical analysis and applications Jg. 519; H. 1; S. 126705
1. Verfasser: Nikitopoulos, Evangelos A.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Elsevier Inc 01.03.2023
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ISSN:0022-247X, 1096-0813
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Abstract 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.
AbstractList 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.
ArticleNumber 126705
Author Nikitopoulos, Evangelos A.
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  organization: Department of Mathematics, University of California, San Diego, La Jolla, CA 92093-0112, USA
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Issue 1
Keywords Noncommutative Banach function space
Multiple operator integral
Measurable operator
Fréchet derivative
Symmetrically normed ideal
Functional calculus
Language English
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Snippet 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...
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StartPage 126705
SubjectTerms Fréchet derivative
Functional calculus
Measurable operator
Multiple operator integral
Noncommutative Banach function space
Symmetrically normed ideal
Title Higher derivatives of operator functions in ideals of von Neumann algebras
URI https://dx.doi.org/10.1016/j.jmaa.2022.126705
Volume 519
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