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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Vydáno v:Journal of mathematical analysis and applications Ročník 519; číslo 1; s. 126705
Hlavní autor: Nikitopoulos, Evangelos A.
Médium: Journal Article
Jazyk:angličtina
Vydáno: Elsevier Inc 01.03.2023
Témata:
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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Popis
Shrnutí: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