Estimates for the perturbations of the operator function \(D(1+D^ 2)^{-1/2}\), based on a nonlinear functional in symmetric spaces. (Q1432126)

The present paper is devoted to an estimate of the norm of a perturbation \(F(D)-F(D_0)\) of the operator function \(F(D)=D(1+D^2)^{-1/2}\) in terms of the norm of \(D-D_0\), which is assumed to be a bounded operator from some von Neumann algebra \(M\). On the other hand, the operators \(D\) and \(D_0\) may be unbounded and are taken from a noncommutative Banach function space, more specifically, a symmetric space of measurable operators associated with the algebra \(M\). The authors motivate their results by stating some applications to the theory of unbounded Fredholm modules.