2023/02/13 by Ghara, Soumitra, Mashreghi, Javad, Ransford, Thomas
#30H20 #30H45 #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Primary 46A35 #Secondary 30H10
paper · doi:10.48550/arxiv.2302.06720
We formalize the observation that the same summability methods converge in a Banach space X and its dual X^*. At the same time we determine conditions under which these methods converge in the weak and weak*-topologies on X and X^* respectively. We also derive a general limitation theorem, which yields a necessary condition for the convergence of a summability method in X. These results are then illustrated by applications to a wide variety of function spaces, including spaces of continuous functions, Lebesgue spaces, the disk algebra, Hardy and Bergman spaces, the BMOA space, the Bloch space, and de Branges-Rovnyak spaces. Our approach shows that all these applications flow from just two abstract theorems.