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Non-selfadjoint operator functions
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Spectral properties of linear operators and operator functions can be used to analyze models in nature. When dispersion and damping are taken into account, the dependence of the spectral parameter is in general non-linear and the operators are not selfadjoint.

In this thesis non-selfadjoint operator functions are studied and several methods for obtaining properties of unbounded non-selfadjoint operator functions are presented. Equivalence is used to characterize operator functions since two equivalent operators share many significant characteristics such as the spectrum and closeness. Methods of linearization and other types of equivalences are presented for a class of unbounded operator matrix functions.

To study properties of the spectrum for non-selfadjoint operator functions, the numerical range is a powerful tool. The thesis introduces an optimal enclosure of the numerical range of a class of unbounded operator functions. The new enclosure can be computed explicitly, and it is investigated in detail. Many properties of the numerical range such as the number of components can be deduced from the enclosure. Furthermore, it is utilized to prove the existence of an infinite number of eigenvalues accumulating to specific points in the complex plane. Among the results are proofs of accumulation of eigenvalues to the singularities of a class of unbounded rational operator functions. The enclosure of the numerical range is also used to find optimal and computable estimates of the norm of resolvent and a corresponding enclosure of the ε-pseudospectrum. 

Place, publisher, year, edition, pages
Umeå: Umeå universitet , 2017. , p. 21
Series
Research report in mathematics, ISSN 1653-0810 ; 60
Keywords [en]
Non-linear spectral problem, numerical range, pseudospectrum, resolvent estimate, equivalence after extension, block operator matrices, operator functions, operator pencil, spectral divisor, joint numerical range
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-143085ISBN: 978-91-7601-787-6 (print)OAI: oai:DiVA.org:umu-143085DiVA, id: diva2:1166532
Public defence
2018-01-19, MA 121, MIT-huset, Umeå, 09:00 (English)
Opponent
Supervisors
Available from: 2017-12-20 Created: 2017-12-15 Last updated: 2018-06-09Bibliographically approved
List of papers
1. Enclosure of the Numerical Range of a Class of Non-selfadjoint Rational Operator Functions
Open this publication in new window or tab >>Enclosure of the Numerical Range of a Class of Non-selfadjoint Rational Operator Functions
2017 (English)In: Integral equations and operator theory, ISSN 0378-620X, E-ISSN 1420-8989, Vol. 88, no 2, p. 151-184Article in journal (Refereed) Published
Abstract [en]

In this paper we introduce an enclosure of the numerical range of a class of rational operator functions. In contrast to the numerical range the presented enclosure can be computed exactly in the infinite dimensional case as well as in the finite dimensional case. Moreover, the new enclosure is minimal given only the numerical ranges of the operator coefficients and many characteristics of the numerical range can be obtained by investigating the enclosure. We introduce a pseudonumerical range and study an enclosure of this set. This enclosure provides a computable upper bound of the norm of the resolvent.

Keywords
Non-linear spectral problem, Numerical range, Pseudospectra, Resolvent estimate
National Category
Mathematics
Identifiers
urn:nbn:se:umu:diva-137726 (URN)10.1007/s00020-017-2378-6 (DOI)000405016300001 ()
Funder
Swedish Research Council, 621-2012-3863
Available from: 2017-07-07 Created: 2017-07-07 Last updated: 2018-06-09Bibliographically approved
2. On equivalence and linearization of operator matrix functions with unbounded entries
Open this publication in new window or tab >>On equivalence and linearization of operator matrix functions with unbounded entries
2017 (English)In: Integral equations and operator theory, ISSN 0378-620X, E-ISSN 1420-8989, Vol. 89, no 4, p. 465-492Article in journal (Refereed) Published
Abstract [en]

In this paper we present equivalence results for several types of unbounded operator functions. A generalization of the concept equivalence after extension is introduced and used to prove equivalence and linearization for classes of unbounded operator functions. Further, we deduce methods of finding equivalences to operator matrix functions that utilizes equivalences of the entries. Finally, a method of finding equivalences and linearizations to a general case of operator matrix polynomials is presented.

Keywords
Equivalence after extension, Block operator matrices, Operator functions, Spectrum
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-142058 (URN)10.1007/s00020-017-2415-5 (DOI)000416537600001 ()
Available from: 2017-11-17 Created: 2017-11-17 Last updated: 2018-06-09Bibliographically approved
3. Accumulation of Complex Eigenvalues of a Class of Analytic Operator Functions
Open this publication in new window or tab >>Accumulation of Complex Eigenvalues of a Class of Analytic Operator Functions
(English)Manuscript (preprint) (Other academic)
Keywords
Operator pencil, spectral divisor, numerical range, non-linear spectral problem
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-143083 (URN)
Available from: 2017-12-15 Created: 2017-12-15 Last updated: 2018-06-09
4. Enclosure of the Numerical Range and Resolvent Estimates of Non-Selfadjoint Operator Functions
Open this publication in new window or tab >>Enclosure of the Numerical Range and Resolvent Estimates of Non-Selfadjoint Operator Functions
(English)Manuscript (preprint) (Other academic)
Keywords
Non-linear spectral problem, numerical range, joint numerical range, pseudospectrum, resolvent estimate
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-143084 (URN)
Available from: 2017-12-15 Created: 2017-12-15 Last updated: 2018-06-09

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