The Asymptotic Behaviour of Semigroups of Linear Operators - Couverture souple

Neerven, Jan Van

Couverture souple

ISBN 10 : 3034899440 ISBN 13 : 9783034899444

Editeur : Birkhäuser, 2011

Synopsis

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.

Détails de l'édition

• Éditeur: Birkhäuser
• Date d'édition: 2011
• Langue: anglais
• ISBN 10: 3034899440
• ISBN 13: 9783034899444
• Reliure: Broché
• Nombre de pages: 256
• Coordonnées du fabricant: non disponible