Topics in complex analysis and function spaces

There are three parts of this work. In the first part we give an explicit computation of the Banach envelope for non-locally convex Paley-Wiener type spaces.; In the second part we develop more general tools for the treatment of harmonic and analytic functions which take values in a quasi-Banach space. We are able to give a description of boundary value functions for the classes . Our results extend the well known case of scalar valued functions on the disc and the half plane.; The third part is joint work with Nigel Kalton and Tamara Kucherenko. It is shown that R-bounded and weakly compact semigroups on L₁ and C(K) can only exist for ℓ₁ and c₀. More generally, R-bounded weakly compact commuting approximating sequences in Banach spaces are discussed.