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
Author
Summary, in English
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.
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.
Department/s
Publishing year
2007
Language
English
Document type
Dissertation
Publisher
Centre for Mathematical Sciences, Lund University
Topic
- Mathematics
Keywords
- Qp-spaces
- Non-tangential limits
- Shift operator
- Mathematics
- Matematik
Status
Published
Supervisor
ISBN/ISSN/Other
- ISBN: 978-91-628-7270-0
Defence date
17 October 2007
Defence time
10:15
Defence place
Sal C, Matematikcentrum
Opponent
- Hari Bercovici (Professor)