Operator differentiable functions and derivations of operator algebras
type: Item
instance of: scholarly article
title: Operator differentiable functions and derivations of operator algebras
zbMATH Open document ID: 0888.46020
DOI: 10.1007/BF02509624
publication date: 1997-11-05T00:00:00Z
review text: Let \(H\) be a separable Hilbert space, \(I\) a real (open or not) interval, \(B(H)\) the set of all bounded operators and \(B_I(H)\) the set of all hermitian elements \(T\) in \(B(H)\) such that the spectrum of \(T\) is contained in \(I\). A continuous complex valued function \(f:I\to\mathbb{C}\) defines a map \(f_\sigma:B_I(H)\to B(H)\) by \(T\to f(T)\). \(f\) is called operator differentiable on \(I\) if \(f_\sigma\) is Gâteaux or Fréchet differentiable on \(B_I(H)\). Among several results mentioned without proofs, it is stated that \(f\in C(I)\) is Gâteaux differentiable if and only if it is Fréchet differentiable. A similar result is also stated when \(f\) is operator monotone.
Mathematics Subject Classification ID: 46H05
Mathematics Subject Classification ID: 46G05
Mathematics Subject Classification ID: 46H30
zbMATH DE Number: 1053626
zbMATH Keywords: derivations
zbMATH Keywords: multipliers
zbMATH Keywords: compact operators
zbMATH Keywords: operator differentiable
zbMATH Keywords: Gâteaux or Fréchet differentiable
zbMATH Keywords: operator monotone
author: E. Kissin
author: V. S. Shulman
reviewed by: Taqdir Husain
MaRDI profile type: Publication
cites work: Operator differentiable functions
cites work: Q5686515
full work available at URL: https://doi.org/10.1007/bf02509624
OpenAlex ID: W1963734930
published in: Functional Analysis and Its Applications
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description: scientific article; zbMATH DE number 1053626