The Asymptotic Behaviour of Semigroups of Linear Operators
Neerven, Jan Van
| Autor: | Neerven, Jan Van |
|---|---|
| Themengebiete: | Linear |
| Veröffentlichungsdatum: | 30.07.1996 |
| EAN: | 9783764354558 |
| Auflage: | 1996 |
| Sprache: | Englisch |
| Seitenzahl: | 256 |
| Produktart: | Gebunden |
| Verlag: | Birkhäuser Basel Springer Basel AG |
Produktinformationen "The Asymptotic Behaviour of Semigroups of Linear Operators"
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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