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Torshage, Axel

Open this publication in new window or tab >>Accumulation of complex eigenvalues of a class of analytic operator functions### Engström, Christian

### Torshage, Axel

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_0_j_idt188_some",{id:"formSmash:j_idt184:0:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_0_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_0_j_idt188_otherAuthors",{id:"formSmash:j_idt184:0:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_0_j_idt188_otherAuthors",multiple:true}); 2018 (English)In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 275, no 2, p. 442-477Article in journal (Refereed) Published
##### Abstract [en]

##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:umu:diva-146118 (URN)10.1016/j.jfa.2018.03.019 (DOI)000434877800008 ()2-s2.0-85045081595 (Scopus ID)
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##### Funder

Swedish Research Council, 621-2012-3863
Available from: 2018-04-01 Created: 2018-04-01 Last updated: 2018-09-27Bibliographically approved

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

For analytic operator functions, we prove accumulation of branches of complex eigenvalues to the essential spectrum. Moreover, we show minimality and completeness of the corresponding system of eigenvectors and associated vectors. These results are used to prove sufficient conditions for eigenvalue accumulation to the poles and to infinity of rational operator functions. Finally, an application of electromagnetic field theory is given.

Open this publication in new window or tab >>Enclosure of the Numerical Range of a Class of Non-selfadjoint Rational Operator Functions### Engström, Christian

### Torshage, Axel

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_some",{id:"formSmash:j_idt184:1:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_otherAuthors",{id:"formSmash:j_idt184:1:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_otherAuthors",multiple:true}); 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]

##### 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 ()
#####

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##### Funder

Swedish Research Council, 621-2012-3863
Available from: 2017-07-07 Created: 2017-07-07 Last updated: 2018-06-09Bibliographically approved

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

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.

Open this publication in new window or tab >>Non-selfadjoint operator functions### Torshage, Axel

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_some",{id:"formSmash:j_idt184:2:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_otherAuthors",{id:"formSmash:j_idt184:2:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_otherAuthors",multiple:true}); 2017 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Umeå: Umeå universitet, 2017. p. 21
##### Series

Research report in mathematics, ISSN 1653-0810 ; 60
##### Keywords

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

### Marletta, Marco

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##### Supervisors

### Engström, Christian

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_j_idt365",{id:"formSmash:j_idt184:2:j_idt188:j_idt365",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_j_idt365",multiple:true});
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Available from: 2017-12-20 Created: 2017-12-15 Last updated: 2018-06-09Bibliographically approved

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.

School of Mathematics, Cardiff University, United Kingdom.

Open this publication in new window or tab >>On equivalence and linearization of operator matrix functions with unbounded entries### Engström, Christian

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.### Torshage, Axel

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_some",{id:"formSmash:j_idt184:3:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_otherAuthors",{id:"formSmash:j_idt184:3:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_otherAuthors",multiple:true}); 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]

##### 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 ()
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Available from: 2017-11-17 Created: 2017-11-17 Last updated: 2018-06-09Bibliographically approved

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.

Open this publication in new window or tab >>Accumulation of Complex Eigenvalues of a Class of Analytic Operator Functions### Engström, Christian

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.### Torshage, Axel

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_some",{id:"formSmash:j_idt184:4:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_otherAuthors",{id:"formSmash:j_idt184:4:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_otherAuthors",multiple:true}); (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)
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Available from: 2017-12-15 Created: 2017-12-15 Last updated: 2018-06-09

Open this publication in new window or tab >>Enclosure of the Numerical Range and Resolvent Estimates of Non-Selfadjoint Operator Functions### Torshage, Axel

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_5_j_idt188_some",{id:"formSmash:j_idt184:5:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_5_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_5_j_idt188_otherAuthors",{id:"formSmash:j_idt184:5:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_5_j_idt188_otherAuthors",multiple:true}); (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)
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Available from: 2017-12-15 Created: 2017-12-15 Last updated: 2018-06-09