An application of spectral calculus to the problem of saturation in approximation theory

Let , be a net of bounded linear operators on the Banach space E converging strongly to the identity on E. For a given complex-valued function f of a fixed type we consider the net . Among other things we shall show that under reasonable conditions the saturation space of with respect to a given net of positive real numbers converging to zero is equal to that one of . More generally we consider nets where is a net of complex-valued functions and we determine the saturation space of such a net in dependence of the saturation space of