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
(2007)- 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] ... (More) - 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. (Less) - Abstract (Swedish)
- Popular Abstract in Swedish
Del I - "The Shift Operator on Spaces of Vector-valued Analytic Functions" består av tre artiklar som alla handlar om en viss sorts operatorer i Cowen-Douglas klassen med spektrum på enhetsskivan D, eller om man så vill, om operatorn Mz (multiplikation med z) på Hilbert rum H av vektorvärda analytiska funktioner på D. Den första artikeln, "On the Cowen-Douglas class for Banach space operators", tjänar som en introduktion till de senare två artiklarna. I denna ges ett elementärt bevis av sambandet mellan operatorer i Cowen-Douglas klassen och Mz på Hilbertrum H av analytiska funktioner. Den andra artikeln, "Boundary behavior in Hilbert spaces of vector-valued analytic functions" [Journal of... (More) - Popular Abstract in Swedish
Del I - "The Shift Operator on Spaces of Vector-valued Analytic Functions" består av tre artiklar som alla handlar om en viss sorts operatorer i Cowen-Douglas klassen med spektrum på enhetsskivan D, eller om man så vill, om operatorn Mz (multiplikation med z) på Hilbert rum H av vektorvärda analytiska funktioner på D. Den första artikeln, "On the Cowen-Douglas class for Banach space operators", tjänar som en introduktion till de senare två artiklarna. I denna ges ett elementärt bevis av sambandet mellan operatorer i Cowen-Douglas klassen och Mz på Hilbertrum H av analytiska funktioner. Den andra artikeln, "Boundary behavior in Hilbert spaces of vector-valued analytic functions" [Journal of Functional Analysis 247, 2007, p. 169-201], handlar främst om att visa att funktionerna i H har icketangentiella gränsvärden som en direkt följd av diverse operatorteoretiska antaganden på Mz. I den tredje artikeln "On the index in Hilbert spaces of vector-valued analytic functions" använder vi sedan dessa resultat för att härleda operatorteoretiska egenskaper hos Mz, och vi besvarar även de frågor som lämnades öppna i den andra artikeln. Dessa artiklar utvidgar resultat av Alexandru Aleman, Stefan Richter och Carl Sundberg som endast gäller fallet då H består av komplexvärda analytiska funktioner.
Del II består endast av artikeln "Fatou-type theorems for general approximate identities" [Mathematica Scandinavica, to appear]. Där generaliseras Fatou's välkända sats om konvergensområden för konvolutionen av en funktion med Poissonkärnan till att gälla för en stor klass av approximativa enheter. Huvudresultatet i denna artikel säger visar att dessa regioner ibland är större än de klassiska icke-tangentiella områdena.
Slutligen, i Del III återfinns de två artiklarna "Preduals of Qp-spaces" [Complex Variables and Elliptic Equations, Vol 52, Issue 7, 2007, p. 605-628] samt "Preduals of Qp-spaces II - Carleson imbeddings and atomic decompositions" [Complex Variables and Elliptic Equations, Vol 52, Issue 7, 2007, p. 629-653], som är ett gemensamt samarbete med Anna-Maria Persson och Alexandru Aleman. Vi utvidgar i dessa Fefferman's dualitetssats till Qp-rum och utforskar diverse konsekvenser. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/598895
- author
- Carlsson, Marcus LU
- supervisor
- opponent
-
- Professor Bercovici, Hari, Indiana University, USA
- organization
- publishing date
- 2007
- type
- Thesis
- publication status
- published
- subject
- keywords
- Qp-spaces, Non-tangential limits, Shift operator, Mathematics, Matematik
- pages
- 147 pages
- publisher
- Centre for Mathematical Sciences, Lund University
- defense location
- Sal C, Matematikcentrum
- defense date
- 2007-10-17 10:15:00
- ISBN
- 978-91-628-7270-0
- language
- English
- LU publication?
- yes
- id
- 1d11c69a-3a85-4f19-8c70-8ede2ab98c98 (old id 598895)
- date added to LUP
- 2016-04-01 16:02:10
- date last changed
- 2018-11-21 20:38:14
@phdthesis{1d11c69a-3a85-4f19-8c70-8ede2ab98c98, abstract = {{This thesis consists of six articles on three different subjects<br/><br> <br/><br> in the area of complex analysis, operator theory and harmonic<br/><br> <br/><br> analysis.<br/><br> <br/><br> Part I - "The Shift Operator on Spaces of Vector-valued Analytic<br/><br> <br/><br> Functions" consists of three closely connected articles that<br/><br> <br/><br> investigate certain operators in the Cowen-Douglas class with<br/><br> <br/><br> spectrum D - the unit disc, or equivalently, the shift operator<br/><br> <br/><br> M_z (multiplication by $z$) on Hilbert spaces of vector-valued<br/><br> <br/><br> analytic functions on D. The first article "On the<br/><br> <br/><br> Cowen-Douglas class for Banach space operators" [submitted] serves<br/><br> <br/><br> as an introduction and establishes the (well-known) connection<br/><br> <br/><br> between Cowen-Douglas operators and M_z on spaces H of<br/><br> <br/><br> vector-valued analytic functions. The second article<br/><br> <br/><br> "Boundary behavior in Hilbert spaces of vector-valued<br/><br> <br/><br> analytic functions" [Journal of Functional Analysis 247, 2007, p.<br/><br> <br/><br> 169-201], is mainly concerned with proving that the functions in<br/><br> <br/><br> H have a controlled boundary behavior under various<br/><br> <br/><br> operator-theoretic assumptions on M_z. In the third article,<br/><br> <br/><br> "On the index in Hilbert spaces of vector-valued analytic<br/><br> <br/><br> functions" [submitted], we then use the results from the second<br/><br> <br/><br> article to deduce properties of the operator M_z, and we also<br/><br> <br/><br> resolve the main questions left open in the second article. These<br/><br> <br/><br> articles extend results by Alexandru Aleman, Stefan Richter and Carl<br/><br> <br/><br> Sundberg concerning the case when H consists of C-valued<br/><br> <br/><br> analytic functions.<br/><br> <br/><br> Part II consists of a single article - "Fatou-type<br/><br> <br/><br> theorems for general approximate identities" [Mathematica<br/><br> <br/><br> Scandinavica, to appear]. It generalizes Fatou's well known<br/><br> <br/><br> theorem about convergence regions for the convolution of a<br/><br> <br/><br> function with the Poisson kernel, in the sense that I consider any<br/><br> <br/><br> approximate identity subject to quite loose assumptions. The main<br/><br> <br/><br> theorem shows that the corresponding convergence regions are<br/><br> <br/><br> sometimes effectively larger than the non-tangential ones.<br/><br> <br/><br> Finally, in Part III we have the articles "Preduals of<br/><br> <br/><br> Q_p-spaces" [Complex Variables and Elliptic Equations, Vol 52,<br/><br> <br/><br> Issue 7, 2007, p. 605-628] and "Preduals of Q_p-spaces<br/><br> <br/><br> II - Carleson imbeddings and atomic decompositions" [Complex<br/><br> <br/><br> Variables and Elliptic Equations, Vol 52, Issue 7, 2007, p.<br/><br> <br/><br> 629-653], which are a joint work with Anna-Maria Persson and<br/><br> <br/><br> Alexandru Aleman. We extend the Fefferman duality theorem to the<br/><br> <br/><br> recently introduced Q_p-spaces and explore some of its<br/><br> <br/><br> consequences.}}, author = {{Carlsson, Marcus}}, isbn = {{978-91-628-7270-0}}, keywords = {{Qp-spaces; Non-tangential limits; Shift operator; Mathematics; Matematik}}, language = {{eng}}, publisher = {{Centre for Mathematical Sciences, Lund University}}, school = {{Lund University}}, title = {{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}}, year = {{2007}}, }