Classes of operator-smooth functions. II: Operator-differentiable functions (Q1882431)

The paper continues a study of operator-smooth functions of a real variable started in [Proc. Edinb. Math. Soc., II. Ser. 48, 151--173 (2005; Zbl 1065.47010)] and deals with the spaces of Gâteaux and Fréchet operator differentiable functions. The authors introduce the notion of Gâteaux and Fréchet \(J\)-differentiable functions for symmetrically normed ideals \(J\) from the space \(B(H)\) of all linear bounded operators on a Hilbert space \(H\) and show that, for separable \(J\), the space of Gâteaux \(J\)-differentiable functions consists of all \(J\)-Lipschitz differentiable functions, whereas for \(J = B(H)\) this is not true. Besides, the operator Lipschitz functions are exactly those functions which are Gâteaux differentiable along ``compact'' directions. Then, in particular, the authors obtain the following results. A function is Fréchet \(J\)-differentiable if it is Fréchet \(J\)-differentiable along ``finite rank'' directions. For \(J = B(H)\), the notions of the Gâteaux and Fréchet \(J\)-differentiability of functions coincide. If \(H\) is finite-dimensional, then the spaces of Gâteaux and Fréchet operator differentiable functions coincide. The Gâteaux \(J\)-differentiable functions constitute a closed subspace in the space of all \(J\)-Lipschitz functions and both spaces are non-separable. The subspace of all Fréchet \(J\)-differentiable functions is closed in the space of all \(J\)-Lipschitz functions and is separable for \(J = B(H)\). The space of functions acting on the domains of all weakly closed \(*\)-derivatives of \(C^*\)-algebras coincides with the space of all operator Lipschitz functions. One can see some other interesting results and open problems in the paper. [The third part of this series has appeared in Proc. Edinb. Math. Soc., II. Ser. 48, 175--197 (2005; Zbl 1065.47011).]