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Quotients of Reproducing Kernels : Applications in Complex Analysis and Operator Theory

Weiström Dahlin, Frej LU (2026) In Doctoral Theses in Mathematical Sciences 2026(2).
Abstract
This thesis studies reproducing kernels that are realized as pointwise quotients of two other kernels, including far-
reaching generalizations of de Branges–Rovnyak spaces. In the first article, we study reproducing kernels arising
from the well-known multiplier criterion. They are intimately connected to certain operator inequalities, such as
the famous inequality of Shimorin in sub-Bergman spaces, which extend Sarason’s sub-Hardy spaces. We develop
a model reminiscent of the Sz.-Nagy–Foiaş model. As an application we resolve a conjecture regarding the density
of polynomials in certain classes of weighted sub-Bergman spaces. In the second article we generalize the classical
Julia–Carathéodory theorem via... (More)
This thesis studies reproducing kernels that are realized as pointwise quotients of two other kernels, including far-
reaching generalizations of de Branges–Rovnyak spaces. In the first article, we study reproducing kernels arising
from the well-known multiplier criterion. They are intimately connected to certain operator inequalities, such as
the famous inequality of Shimorin in sub-Bergman spaces, which extend Sarason’s sub-Hardy spaces. We develop
a model reminiscent of the Sz.-Nagy–Foiaş model. As an application we resolve a conjecture regarding the density
of polynomials in certain classes of weighted sub-Bergman spaces. In the second article we generalize the classical
Julia–Carathéodory theorem via reproducing kernels. We develop a new boundary notion and approach regions
to it, entirely in terms of reproducing kernels. We also introduce composition factors as a kernel-theoretic alternative
to analytic selfmaps. In the third article we identify co-isometric weighted composition operators as composition
factors. Moreover, we extend results of Mas, Martín, and Vukotić from the unit disk to the polydisk. Specifically,
under mild regularity assumptions on a reproducing kernel 𝑘 on the polydisk, we prove a dichotomy for rank 1
composition factors. The set is either all analytic automorphisms of the polydisk, in which case 𝑘 is a positive power
of the Szegő kernel, or exactly the rotations composed with a permutation. (Less)
Please use this url to cite or link to this publication:
author
supervisor
opponent
  • Professor Jury, Michael, University of Florida, Gainesville, Florida, US
organization
publishing date
type
Thesis
publication status
published
subject
keywords
Reproducing kernels, de Branges–Rovnyak spaces, Complex analysis, Operator theory
in
Doctoral Theses in Mathematical Sciences
volume
2026
issue
2
pages
112 pages
publisher
Mathematics Centre for Mathematical Sciences Lund University Lund
defense location
Hörmandersalen (M:H), Matematikhuset.
defense date
2026-03-20 13:00:00
ISBN
978-91-8104-864-3
978-91-8104-865-0
language
English
LU publication?
yes
id
a2193a0e-d0cf-4ac4-bded-e608e38ea8a2
date added to LUP
2026-02-23 22:51:44
date last changed
2026-02-24 14:02:05
@phdthesis{a2193a0e-d0cf-4ac4-bded-e608e38ea8a2,
  abstract     = {{This thesis studies reproducing kernels that are realized as pointwise quotients of two other kernels, including far-<br/>reaching generalizations of de Branges–Rovnyak spaces. In the first article, we study reproducing kernels arising<br/>from the well-known multiplier criterion. They are intimately connected to certain operator inequalities, such as<br/>the famous inequality of Shimorin in sub-Bergman spaces, which extend Sarason’s sub-Hardy spaces. We develop<br/>a model reminiscent of the Sz.-Nagy–Foiaş model. As an application we resolve a conjecture regarding the density<br/>of polynomials in certain classes of weighted sub-Bergman spaces. In the second article we generalize the classical<br/>Julia–Carathéodory theorem via reproducing kernels. We develop a new boundary notion and approach regions<br/>to it, entirely in terms of reproducing kernels. We also introduce composition factors as a kernel-theoretic alternative<br/>to analytic selfmaps. In the third article we identify co-isometric weighted composition operators as composition<br/>factors. Moreover, we extend results of Mas, Martín, and Vukotić from the unit disk to the polydisk. Specifically,<br/>under mild regularity assumptions on a reproducing kernel 𝑘 on the polydisk, we prove a dichotomy for rank 1<br/>composition factors. The set is either all analytic automorphisms of the polydisk, in which case 𝑘 is a positive power<br/>of the Szegő kernel, or exactly the rotations composed with a permutation.}},
  author       = {{Weiström Dahlin, Frej}},
  isbn         = {{978-91-8104-864-3}},
  keywords     = {{Reproducing kernels; de Branges–Rovnyak spaces; Complex analysis; Operator theory}},
  language     = {{eng}},
  month        = {{02}},
  number       = {{2}},
  publisher    = {{Mathematics Centre for Mathematical Sciences Lund University Lund}},
  school       = {{Lund University}},
  series       = {{Doctoral Theses in Mathematical Sciences}},
  title        = {{Quotients of Reproducing Kernels : Applications in Complex Analysis and Operator Theory}},
  url          = {{https://lup.lub.lu.se/search/files/243085066/Avhandling_Frej_Dahlin_LUCRIS.pdf}},
  volume       = {{2026}},
  year         = {{2026}},
}