| 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 |
| Authors | Marcus Carlsson |
| Full-text | Available as PDF, Restricted Access |
| Year | 2007 |
| Pages | 147 |
| Document type | Thesis |
| Language | eng |
| Publisher | Centre for Mathematical Sciences, Lund University. |
| Abstract English | This thesis consists of six articles on three different subjects<br> <br> in the area of complex analysis, operator theory and harmonic<br> <br> analysis.<br> <br> Part I - "The Shift Operator on Spaces of Vector-valued Analytic<br> <br> Functions" consists of three closely connected articles that<br> <br> investigate certain operators in the Cowen-Douglas class with<br> <br> spectrum D - the unit disc, or equivalently, the shift operator<br> <br> M_z (multiplication by $z$) on Hilbert spaces of vector-valued<br> <br> analytic functions on D. The first article "On the<br> <br> Cowen-Douglas class for Banach space operators" submitted serves<br> <br> as an introduction and establishes the (well-known) connection<br> <br> between Cowen-Douglas operators and M_z on spaces H of<br> <br> vector-valued analytic functions. The second article<br> <br> "Boundary behavior in Hilbert spaces of vector-valued<br> <br> analytic functions" Journal of Functional Analysis 247, 2007, p.<br> <br> 169-201, is mainly concerned with proving that the functions in<br> <br> H have a controlled boundary behavior under various<br> <br> operator-theoretic assumptions on M_z. In the third article,<br> <br> "On the index in Hilbert spaces of vector-valued analytic<br> <br> functions" submitted, we then use the results from the second<br> <br> article to deduce properties of the operator M_z, and we also<br> <br> resolve the main questions left open in the second article. These<br> <br> articles extend results by Alexandru Aleman, Stefan Richter and Carl<br> <br> Sundberg concerning the case when H consists of C-valued<br> <br> analytic functions.<br> <br> Part II consists of a single article - "Fatou-type<br> <br> theorems for general approximate identities" Mathematica<br> <br> Scandinavica, to appear. It generalizes Fatou's well known<br> <br> theorem about convergence regions for the convolution of a<br> <br> function with the Poisson kernel, in the sense that I consider any<br> <br> approximate identity subject to quite loose assumptions. The main<br> <br> theorem shows that the corresponding convergence regions are<br> <br> sometimes effectively larger than the non-tangential ones.<br> <br> Finally, in Part III we have the articles "Preduals of<br> <br> Q_p-spaces" Complex Variables and Elliptic Equations, Vol 52,<br> <br> Issue 7, 2007, p. 605-628 and "Preduals of Q_p-spaces<br> <br> II - Carleson imbeddings and atomic decompositions" Complex<br> <br> Variables and Elliptic Equations, Vol 52, Issue 7, 2007, p.<br> <br> 629-653, which are a joint work with Anna-Maria Persson and<br> <br> Alexandru Aleman. We extend the Fefferman duality theorem to the<br> <br> recently introduced Q_p-spaces and explore some of its<br> <br> consequences. |
| Abstract Swedish | This thesis consists of six articles on three different subjects<br> <br> in the area of complex analysis, operator theory and harmonic<br> <br> analysis.<br> <br> Part I - "The Shift Operator on Spaces of Vector-valued Analytic<br> <br> Functions" consists of three closely connected articles that<br> <br> investigate certain operators in the Cowen-Douglas class with<br> <br> spectrum D - the unit disc, or equivalently, the shift operator<br> <br> M_z (multiplication by $z$) on Hilbert spaces of vector-valued<br> <br> analytic functions on D. The first article "On the<br> <br> Cowen-Douglas class for Banach space operators" submitted serves<br> <br> as an introduction and establishes the (well-known) connection<br> <br> between Cowen-Douglas operators and M_z on spaces H of<br> <br> vector-valued analytic functions. The second article<br> <br> "Boundary behavior in Hilbert spaces of vector-valued<br> <br> analytic functions" Journal of Functional Analysis 247, 2007, p.<br> <br> 169-201, is mainly concerned with proving that the functions in<br> <br> H have a controlled boundary behavior under various<br> <br> operator-theoretic assumptions on M_z. In the third article,<br> <br> "On the index in Hilbert spaces of vector-valued analytic<br> <br> functions" submitted, we then use the results from the second<br> <br> article to deduce properties of the operator M_z, and we also<br> <br> resolve the main questions left open in the second article. These<br> <br> articles extend results by Alexandru Aleman, Stefan Richter and Carl<br> <br> Sundberg concerning the case when H consists of C-valued<br> <br> analytic functions.<br> <br> Part II consists of a single article - "Fatou-type<br> <br> theorems for general approximate identities" Mathematica<br> <br> Scandinavica, to appear. It generalizes Fatou's well known<br> <br> theorem about convergence regions for the convolution of a<br> <br> function with the Poisson kernel, in the sense that I consider any<br> <br> approximate identity subject to quite loose assumptions. The main<br> <br> theorem shows that the corresponding convergence regions are<br> <br> sometimes effectively larger than the non-tangential ones.<br> <br> Finally, in Part III we have the articles "Preduals of<br> <br> Q_p-spaces" Complex Variables and Elliptic Equations, Vol 52,<br> <br> Issue 7, 2007, p. 605-628 and "Preduals of Q_p-spaces<br> <br> II - Carleson imbeddings and atomic decompositions" Complex<br> <br> Variables and Elliptic Equations, Vol 52, Issue 7, 2007, p.<br> <br> 629-653, which are a joint work with Anna-Maria Persson and<br> <br> Alexandru Aleman. We extend the Fefferman duality theorem to the<br> <br> recently introduced Q_p-spaces and explore some of its<br> <br> consequences. |
| Keywords | Qp-spaces, Non-tangential limits, Shift operator, Mathematics, Matematik, |
| ISBN/ISSN/Other | ISSN: 1404-0034 ISBN: 978-91-628-7270-0 |
Questions: webmaster
Last update: 2013-04-11
Centre for Mathematical Sciences, Box 118, SE-22100, Lund. Telefon: +46 46-222 00 00 (vx)