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

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.