The Asymptotic Behaviour of Semigroups of Linear Operators:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Basel
Birkhäuser Basel
1996
|
| Schriftenreihe: | Operator Theory: Advances and Applications
88 |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Beschreibung: | Over the past ten years, the asymptotic theory of one-parameter semigroups of operators has witnessed an explosive development. A number oflong-standing open problems have recently been solved and the theory seems to have obtained a certain degree of maturity. These notes, based on a course delivered at the University of Tiibingen in the academic year 1994-1995, represent a first attempt to organize the available material, most of which exists only in the form of research papers. If A is a bounded linear operator on a complex Banach space X, then it is an easy consequence of the spectral mapping theorem exp(tO"(A)) = O"(exp(tA)), t E JR, and Gelfand's formula for the spectral radius that the uniform growth bound of the wt family {exp(tA)h~o, i. e. the infimum of all wE JR such that II exp(tA)II :::: Me for some constant M and all t 2: 0, is equal to the spectral bound s(A) = sup{Re A : A E O"(A)} of A. This fact is known as Lyapunov's theorem. Its importance resides in the fact that the solutions of the initial value problem du(t) =A () dt u t , u(O) = x, are given by u(t) = exp(tA)x. Thus, Lyapunov's theorem implies that the expo nential growth of the solutions of the initial value problem associated to a bounded operator A is determined by the location of the spectrum of A. |
| Beschreibung: | 1 Online-Ressource (XII, 241 p) |
| ISBN: | 9783034892063 9783034899444 |
| DOI: | 10.1007/978-3-0348-9206-3 |
Internformat
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| 490 | 0 | |a Operator Theory: Advances and Applications |v 88 | |
| 500 | |a Over the past ten years, the asymptotic theory of one-parameter semigroups of operators has witnessed an explosive development. A number oflong-standing open problems have recently been solved and the theory seems to have obtained a certain degree of maturity. These notes, based on a course delivered at the University of Tiibingen in the academic year 1994-1995, represent a first attempt to organize the available material, most of which exists only in the form of research papers. If A is a bounded linear operator on a complex Banach space X, then it is an easy consequence of the spectral mapping theorem exp(tO"(A)) = O"(exp(tA)), t E JR, and Gelfand's formula for the spectral radius that the uniform growth bound of the wt family {exp(tA)h~o, i. e. the infimum of all wE JR such that II exp(tA)II :::: Me for some constant M and all t 2: 0, is equal to the spectral bound s(A) = sup{Re A : A E O"(A)} of A. This fact is known as Lyapunov's theorem. Its importance resides in the fact that the solutions of the initial value problem du(t) =A () dt u t , u(O) = x, are given by u(t) = exp(tA)x. Thus, Lyapunov's theorem implies that the expo nential growth of the solutions of the initial value problem associated to a bounded operator A is determined by the location of the spectrum of A. | ||
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Datensatz im Suchindex
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| isbn | 9783034892063 9783034899444 |
| language | English |
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| spelling | Neerven, Jan Verfasser aut The Asymptotic Behaviour of Semigroups of Linear Operators by Jan Neerven Basel Birkhäuser Basel 1996 1 Online-Ressource (XII, 241 p) txt rdacontent c rdamedia cr rdacarrier Operator Theory: Advances and Applications 88 Over the past ten years, the asymptotic theory of one-parameter semigroups of operators has witnessed an explosive development. A number oflong-standing open problems have recently been solved and the theory seems to have obtained a certain degree of maturity. These notes, based on a course delivered at the University of Tiibingen in the academic year 1994-1995, represent a first attempt to organize the available material, most of which exists only in the form of research papers. If A is a bounded linear operator on a complex Banach space X, then it is an easy consequence of the spectral mapping theorem exp(tO"(A)) = O"(exp(tA)), t E JR, and Gelfand's formula for the spectral radius that the uniform growth bound of the wt family {exp(tA)h~o, i. e. the infimum of all wE JR such that II exp(tA)II :::: Me for some constant M and all t 2: 0, is equal to the spectral bound s(A) = sup{Re A : A E O"(A)} of A. This fact is known as Lyapunov's theorem. Its importance resides in the fact that the solutions of the initial value problem du(t) =A () dt u t , u(O) = x, are given by u(t) = exp(tA)x. Thus, Lyapunov's theorem implies that the expo nential growth of the solutions of the initial value problem associated to a bounded operator A is determined by the location of the spectrum of A. Mathematics Functional analysis Operator theory Operator Theory Functional Analysis Mathematik Banach-Raum (DE-588)4004402-6 gnd rswk-swf Asymptotische Darstellung (DE-588)4003339-9 gnd rswk-swf Halbgruppe (DE-588)4022990-7 gnd rswk-swf Linearer Operator (DE-588)4167721-3 gnd rswk-swf Banach-Raum (DE-588)4004402-6 s Linearer Operator (DE-588)4167721-3 s Halbgruppe (DE-588)4022990-7 s Asymptotische Darstellung (DE-588)4003339-9 s 1\p DE-604 https://doi.org/10.1007/978-3-0348-9206-3 Verlag Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Neerven, Jan The Asymptotic Behaviour of Semigroups of Linear Operators Mathematics Functional analysis Operator theory Operator Theory Functional Analysis Mathematik Banach-Raum (DE-588)4004402-6 gnd Asymptotische Darstellung (DE-588)4003339-9 gnd Halbgruppe (DE-588)4022990-7 gnd Linearer Operator (DE-588)4167721-3 gnd |
| subject_GND | (DE-588)4004402-6 (DE-588)4003339-9 (DE-588)4022990-7 (DE-588)4167721-3 |
| title | The Asymptotic Behaviour of Semigroups of Linear Operators |
| title_auth | The Asymptotic Behaviour of Semigroups of Linear Operators |
| title_exact_search | The Asymptotic Behaviour of Semigroups of Linear Operators |
| title_full | The Asymptotic Behaviour of Semigroups of Linear Operators by Jan Neerven |
| title_fullStr | The Asymptotic Behaviour of Semigroups of Linear Operators by Jan Neerven |
| title_full_unstemmed | The Asymptotic Behaviour of Semigroups of Linear Operators by Jan Neerven |
| title_short | The Asymptotic Behaviour of Semigroups of Linear Operators |
| title_sort | the asymptotic behaviour of semigroups of linear operators |
| topic | Mathematics Functional analysis Operator theory Operator Theory Functional Analysis Mathematik Banach-Raum (DE-588)4004402-6 gnd Asymptotische Darstellung (DE-588)4003339-9 gnd Halbgruppe (DE-588)4022990-7 gnd Linearer Operator (DE-588)4167721-3 gnd |
| topic_facet | Mathematics Functional analysis Operator theory Operator Theory Functional Analysis Mathematik Banach-Raum Asymptotische Darstellung Halbgruppe Linearer Operator |
| url | https://doi.org/10.1007/978-3-0348-9206-3 |
| work_keys_str_mv | AT neervenjan theasymptoticbehaviourofsemigroupsoflinearoperators |