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


Let \mathcal L= (L_𝛼)_𝛼∈ 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(L𝛼))_𝛼. Among other things we shall show that under reasonable conditions the saturation space of with respect to a given net \Phi = (\Phi_𝛼) of positive real numbers converging to zero is equal to that one of f (\mathcal L) . More generally we consider nets ( f_𝛼( L_𝛼)) where (f_𝛼) is a net of complex-valued functions and we determine the saturation space of such a net in dependence of the saturation space of .

DOI Code: 10.1285/i15900932v12p291

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