Operator differentiable functions and derivations of operator algebras
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.
scientific article; zbMATH DE number 1053626
scientific article