CLASSES OF OPERATOR-SMOOTH FUNCTIONS. I. OPERATOR-LIPSCHITZ FUNCTIONS

In this paper we study the spaces of operator-Lipschitz functions and the spaces of functions closed to them: commutator bounded. Apart from the standard operator norm on $B(H)$, we consider a rich variety of symmetric operator norms and spaces of operator-Lipschitz functions with respect to these norms. Our approach is aimed at the investigation of the interrelation and hierarchy of these spaces and of the intrinsic properties of operator-Lipschitz functions.AMS 2000 Mathematics subject classification: Primary 47A56. Secondary 47L20