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.

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 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.

• author: Carlsson, Marcus
• supervisor: Alexandru Aleman
• opponent: Professor Bercovici, Hari, Indiana University, USA
• organization: Mathematics (Faculty of Sciences)
• publishing date: 2007
• type: Thesis
• publication status: published
• subject: Mathematics
• keywords: Qp-spaces, Non-tangential limits, Shift operator, Mathematics, Matematik
• pages: 147 pages
• 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