Operator differentiable functions

A scalar function f is called opertor differentiable if its extension via spectral theory to the self-adjoint members of {Mathematical expression}(H) is differentiable. The study of differentiation and perturbation of such operator functions leads to the theory of mappings defined by the double operator integral {Mathematical expression} We give a new condition under which this mapping is bounded on {Mathematical expression}(H). We also present a means of extending f to a function on all of {Mathematical expression}(H) and determine corresponding perturbation and differentiation formulas. A connection with the "joint Peirce decomposition" from the theory of JB^*-triples is found. As an application we broaden the class of functions known to preserve the domain of the generator of a strongly continuous one-parameter group of^*-automorphisms of a C^*-algebra.