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On equivalence and linearization of operator matrix functions with unbounded entries
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik. (UMIT)
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik. (UMIT)
2017 (engelsk)Inngår i: Integral equations and operator theory, ISSN 0378-620X, E-ISSN 1420-8989, Vol. 89, nr 4, s. 465-492Artikkel i tidsskrift (Fagfellevurdert) 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.

sted, utgiver, år, opplag, sider
2017. Vol. 89, nr 4, s. 465-492
Emneord [en]
Equivalence after extension, Block operator matrices, Operator functions, Spectrum
HSV kategori
Forskningsprogram
matematik
Identifikatorer
URN: urn:nbn:se:umu:diva-142058DOI: 10.1007/s00020-017-2415-5ISI: 000416537600001OAI: oai:DiVA.org:umu-142058DiVA, id: diva2:1158143
Tilgjengelig fra: 2017-11-17 Laget: 2017-11-17 Sist oppdatert: 2018-06-09bibliografisk kontrollert
Inngår i avhandling
1. Non-selfadjoint operator functions
Åpne denne publikasjonen i ny fane eller vindu >>Non-selfadjoint operator functions
2017 (engelsk)Doktoravhandling, med artikler (Annet vitenskapelig)
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. 

sted, utgiver, år, opplag, sider
Umeå: Umeå universitet, 2017. s. 21
Serie
Research report in mathematics, ISSN 1653-0810 ; 60
Emneord
Non-linear spectral problem, numerical range, pseudospectrum, resolvent estimate, equivalence after extension, block operator matrices, operator functions, operator pencil, spectral divisor, joint numerical range
HSV kategori
Forskningsprogram
matematik
Identifikatorer
urn:nbn:se:umu:diva-143085 (URN)978-91-7601-787-6 (ISBN)
Disputas
2018-01-19, MA 121, MIT-huset, Umeå, 09:00 (engelsk)
Opponent
Veileder
Tilgjengelig fra: 2017-12-20 Laget: 2017-12-15 Sist oppdatert: 2018-06-09bibliografisk kontrollert

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