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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 (Engelska)Ingår i: Integral equations and operator theory, ISSN 0378-620X, E-ISSN 1420-8989, Vol. 89, nr 4, s. 465-492Artikel i tidskrift (Refereegranskat) 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.

Ort, förlag, år, upplaga, sidor
2017. Vol. 89, nr 4, s. 465-492
Nyckelord [en]
Equivalence after extension, Block operator matrices, Operator functions, Spectrum
Nationell ämneskategori
Matematisk analys
Forskningsämne
matematik
Identifikatorer
URN: urn:nbn:se:umu:diva-142058DOI: 10.1007/s00020-017-2415-5ISI: 000416537600001OAI: oai:DiVA.org:umu-142058DiVA, id: diva2:1158143
Tillgänglig från: 2017-11-17 Skapad: 2017-11-17 Senast uppdaterad: 2018-06-09Bibliografiskt granskad
Ingår i avhandling
1. Non-selfadjoint operator functions
Öppna denna publikation i ny flik eller fönster >>Non-selfadjoint operator functions
2017 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
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. 

Ort, förlag, år, upplaga, sidor
Umeå: Umeå universitet, 2017. s. 21
Serie
Research report in mathematics, ISSN 1653-0810 ; 60
Nyckelord
Non-linear spectral problem, numerical range, pseudospectrum, resolvent estimate, equivalence after extension, block operator matrices, operator functions, operator pencil, spectral divisor, joint numerical range
Nationell ämneskategori
Matematisk analys
Forskningsämne
matematik
Identifikatorer
urn:nbn:se:umu:diva-143085 (URN)978-91-7601-787-6 (ISBN)
Disputation
2018-01-19, MA 121, MIT-huset, Umeå, 09:00 (Engelska)
Opponent
Handledare
Tillgänglig från: 2017-12-20 Skapad: 2017-12-15 Senast uppdaterad: 2018-06-09Bibliografiskt granskad

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Engström, ChristianTorshage, Axel

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