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Operator models for meromorphic functions of bounded type
Stockholm University, Faculty of Science, Department of Mathematics.ORCID iD: 0009-0005-6275-1269
2025 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of four papers dealing with generalizations of de Branges`s model theory of (cyclic) self-adjoint operators and applications to the extension theory of symmetric operators.

The main project of this PhD thesis, consisting of three papers, is concerned with constructing operator models for meromorphic functions of bounded type. Specifically, it is shown that these functions can be realized as Q-functions of partially fundamentally reducible relations on Krein spaces in a minimal way. The main result can be found in Paper III, while Papers I and II contain related results of a smaller scope.

The main argument of our construction in Paper 3 can be used to generalize the extension theory for symmetric operators with deficiency index (1,1). Specifically, we characterize all one dimensional extensions with non-empty resolvent set  of such an operator via a Krein-type resolvent formula and investigate their spectral properties. This is the content of Paper IV. 

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University , 2025. , p. 46
Keywords [en]
Operator models, Symmetric operators
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-237273ISBN: 978-91-8107-062-0 (print)ISBN: 978-91-8107-063-7 (electronic)OAI: oai:DiVA.org:su-237273DiVA, id: diva2:1921449
Public defence
2025-02-14, lärosal 4, hus 1, Albano, Albanovägen 28, Stockholm, 09:00 (English)
Opponent
Supervisors
Available from: 2025-01-22 Created: 2024-12-16 Last updated: 2025-01-24Bibliographically approved
List of papers
1. Realizations of Meromorphic Functions of Bounded Type
Open this publication in new window or tab >>Realizations of Meromorphic Functions of Bounded Type
2023 (English)In: From Complex Analysis to Operator Theory: A Panorama In Memory of Sergey Naboko / [ed] Malcolm Brown, Fritz Gesztesy, Pavel Kurasov, Ari Laptev, Barry Simon, Gunter Stolz, Ian Wood, Springer, 2023, p. 501-522Chapter in book (Refereed)
Abstract [en]

In this article it is shown that every function meromorphic in the upper halfplane and of bounded type does have a realization with the resolvent of a self-adjoint relation in a Krein space.

Place, publisher, year, edition, pages
Springer, 2023
Series
Operator Theory: Advances and Applications, ISSN 0255-0156, E-ISSN 2296-4878 ; 291
Keywords
Functions of bounded type, Herglotz-Nevanlinna functions, Krein spaces, Quasi-Herglotz functions, Realizations, Self-adjoint relations
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:su:diva-234462 (URN)10.1007/978-3-031-31139-0_18 (DOI)2-s2.0-85160238682 (Scopus ID)978-3-031-31138-3 (ISBN)
Available from: 2024-10-16 Created: 2024-10-16 Last updated: 2024-12-16Bibliographically approved
2. Minimal Realizations of Atomic Density Functions
Open this publication in new window or tab >>Minimal Realizations of Atomic Density Functions
2023 (English)In: Complex Analysis and Operator Theory, ISSN 1661-8254, E-ISSN 1661-8262, Vol. 17, no 5, article id 52Article in journal (Refereed) Published
Abstract [en]

In this article, it is shown that Koebe inner functions and squares of singular Nevanlinna functions, so called atomic density functions, have minimal realizations in reproducing kernel Krein spaces.

Keywords
Realizations, density functions, Herglotz-Nevanlinna functions, Krein spaces
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-213780 (URN)10.1007/s11785-023-01360-w (DOI)000994247100001 ()2-s2.0-85160225994 (Scopus ID)
Available from: 2023-01-17 Created: 2023-01-17 Last updated: 2024-12-16Bibliographically approved
3. Operator models for meromorphic functions of bounded type
Open this publication in new window or tab >>Operator models for meromorphic functions of bounded type
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this article, we construct operator models on Krein spaces for meromorphic functions ofbounded type. This construction is based on certain reproducing kernel Hilbert spaces whichare closely related to model spaces. Specifically, we show that each function of bounded typecorresponds naturally to a pair of such spaces, which extends Helson’s representation theorem.This correspondence enables an explicit construction of our model, where the Krein space is asuitable sum of these identified spaces. Additionally, we establish that the representing self-adjointrelations possess a relatively simple structure, since they turn out to be partially fundamentallyreducible. Conversely, we show that realizations involving such relations correspond to functionsof bounded type.

Keywords
Realizations, Krein space
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-237134 (URN)
Available from: 2024-12-11 Created: 2024-12-11 Last updated: 2024-12-16
4. A generalization  of Krein`s extension formalism for symmetric relations with deficiency index (1,1).
Open this publication in new window or tab >>A generalization  of Krein`s extension formalism for symmetric relations with deficiency index (1,1).
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Let S be a symmetric relation with deficiency index (1, 1). In this article, we extend Krein‘s resolvent formalism in order to describe all, not necessarily self-adjoint, extensions S ⊂ A with nonempty resolvent set. The corresponding Q-functions turn out to be quasi-Herglotz functions. We will use their structure to characterize the spectrum of such extensions. Finally, we also provide amodel for such an extension on a reproducing kernel Hilbert space when S is simple.

Keywords
Symmetric operators, Krein-type formula
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-237136 (URN)
Available from: 2024-12-11 Created: 2024-12-11 Last updated: 2024-12-16

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Emmel, Christian

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