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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

Carlsson, Marcus LU (2007)
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)
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)
Please use this url to cite or link to this publication:
author
supervisor
opponent
  • Professor Bercovici, Hari, Indiana University, USA
organization
publishing date
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
ISSN
1404-0034
ISBN
978-91-628-7270-0
language
English
LU publication?
yes
id
1d11c69a-3a85-4f19-8c70-8ede2ab98c98 (old id 598895)
date added to LUP
2007-11-13 07:22:09
date last changed
2016-09-19 08:44:54
@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},
  issn         = {1404-0034},
  keyword      = {Qp-spaces,Non-tangential limits,Shift operator,Mathematics,Matematik},
  language     = {eng},
  pages        = {147},
  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},
}