The Asymptotic Behaviour of Semigroups of Linear Operators: 88 (Operator Theory: Advances and Applications, 88) - Hardcover

Neerven, Jan Van

Hardcover

ISBN 10: 3764354550 ISBN 13: 9783764354558

Publisher: Birkhäuser, 1996

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.

About this title

Synopsis

The focus of this text is on the relationship between asymptotic behaviour of the semigroup and spectral properties of its infinitesimal generator. Recent developments in the field are covered, such as the Arendt-Batty-Lyubich-Vu theorem, the spectral mapping theorem of Latushkin and Montgomery-Smith, Weis's theorem on stability of positive semigroups in Lp-spaces, the stability theorem for semigroups whose resolvent is bounded in a half-plane, and a systematic theory of individual stability.

Edition details

• Publisher: Birkhäuser
• Publication date: 1996
• Language: English
• ISBN 10: 3764354550
• ISBN 13: 9783764354558
• Binding: Hardcover
• Number of pages: 253