Topics in Complex Analysis and Operator Theory I. The shift operator on spaces of vector-valued analytic functions II. Fatou-type theorems for general approximate identities III. Preduals of Q_p-spaces

Abstract: This thesis consists of six articles on three different subjects in the area of complex analysis, operator theory and harmonic analysis. Part I - "The Shift Operator on Spaces of Vector-valued Analytic Functions" consists of three closely connected articles that investigate certain operators in the Cowen-Douglas class with spectrum D - the unit disc, or equivalently, the shift operator M_z (multiplication by $z$) on Hilbert spaces of vector-valued analytic functions on D. The first article "On the Cowen-Douglas class for Banach space operators" [submitted] serves as an introduction and establishes the (well-known) connection between Cowen-Douglas operators and M_z on spaces H of vector-valued analytic functions. The second article "Boundary behavior in Hilbert spaces of vector-valued analytic functions" [Journal of Functional Analysis 247, 2007, p. 169-201], is mainly concerned with proving that the functions in H have a controlled boundary behavior under various operator-theoretic assumptions on M_z. In the third article, "On the index in Hilbert spaces of vector-valued analytic functions" [submitted], we then use the results from the second article to deduce properties of the operator M_z, and we also resolve the main questions left open in the second article. These articles extend results by Alexandru Aleman, Stefan Richter and Carl Sundberg concerning the case when H consists of C-valued analytic functions. Part II consists of a single article - "Fatou-type theorems for general approximate identities" [Mathematica Scandinavica, to appear]. It generalizes Fatou's well known theorem about convergence regions for the convolution of a function with the Poisson kernel, in the sense that I consider any approximate identity subject to quite loose assumptions. The main theorem shows that the corresponding convergence regions are sometimes effectively larger than the non-tangential ones. Finally, in Part III we have the articles "Preduals of Q_p-spaces" [Complex Variables and Elliptic Equations, Vol 52, Issue 7, 2007, p. 605-628] and "Preduals of Q_p-spaces II - Carleson imbeddings and atomic decompositions" [Complex Variables and Elliptic Equations, Vol 52, Issue 7, 2007, p. 629-653], which are a joint work with Anna-Maria Persson and Alexandru Aleman. We extend the Fefferman duality theorem to the recently introduced Q_p-spaces and explore some of its consequences.