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 |
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01.03.2023
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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. |
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| 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. |
| Author_xml | – sequence: 1 givenname: Evangelos A. surname: Nikitopoulos fullname: Nikitopoulos, Evangelos A. email: enikitop@ucsd.edu organization: Department of Mathematics, University of California, San Diego, La Jolla, CA 92093-0112, USA |
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| Keywords | Noncommutative Banach function space Multiple operator integral Measurable operator Fréchet derivative Symmetrically normed ideal Functional calculus |
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Anal. doi: 10.1016/j.jfa.2005.09.003 – volume: vol. 2250 year: 2019 ident: 10.1016/j.jmaa.2022.126705_br0320 article-title: Multilinear Operator Integrals: Theory and Applications |
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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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| 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 |
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