Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Enclosure of the Numerical Range of a Class of Non-selfadjoint Rational Operator Functions
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. (UMIT)
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
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.

Place, publisher, year, edition, pages
2017. Vol. 88, no 2, p. 151-184
Keyword [en]
Non-linear spectral problem, Numerical range, Pseudospectra, Resolvent estimate
National Category
Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-137726DOI: 10.1007/s00020-017-2378-6ISI: 000405016300001OAI: oai:DiVA.org:umu-137726DiVA, id: diva2:1120953
Funder
Swedish Research Council, 621-2012-3863
Available from: 2017-07-07 Created: 2017-07-07 Last updated: 2017-12-15Bibliographically approved
In thesis
1. Non-selfadjoint operator functions
Open this publication in new window or tab >>Non-selfadjoint operator functions
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
Keyword
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:nbn:se:umu:diva-143085 (URN)978-91-7601-787-6 (ISBN)
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: 2017-12-18Bibliographically approved

Open Access in DiVA

fulltext(4914 kB)19 downloads
File information
File name FULLTEXT01.pdfFile size 4914 kBChecksum SHA-512
2442a562ea651546ec97412b2bd94793c39afabf1d2a7cad1f98f9b39b09d9f8f5e76529cc70f03634046ecf4d3ece4a4ce06c0e0c4483393edbede8973ee590
Type fulltextMimetype application/pdf

Other links

Publisher's full text

Search in DiVA

By author/editor
Engström, ChristianTorshage, Axel
By organisation
Department of Mathematics and Mathematical Statistics
In the same journal
Integral equations and operator theory
Mathematics

Search outside of DiVA

GoogleGoogle Scholar
Total: 19 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

doi
urn-nbn

Altmetric score

doi
urn-nbn
Total: 149 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf