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

Abstract

Let $\mathcal L= (L_?)_?∈ A$, 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 $f (\mathcal L) := ( f(L?))_?$. Among other things we shall show that under reasonable conditions the saturation space of ??? with respect to a given net $\Phi = (\Phi_?)$ of positive real numbers converging to zero is equal to that one of $f (\mathcal L)$ . More generally we consider nets $( f_?( L_?))$ where $(f_?)$ is a net of complex-valued functions and we determine the saturation space of such a net in dependence of the saturation space of ??? .