The Asymptotic Behaviour of Semigroups of Linear Operators (Operator Theory: Advances and Applications)

Neerven, Jan Van

ISBN 10: 3034899440 / ISBN 13: 9783034899444

Verlag: Birkhäuser, 2011

In. Bestandsnummer des Verkäufers ria9783034899444_new

Inhaltsangabe:

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.

Bibliografische Details

Titel: The Asymptotic Behaviour of Semigroups of ...
Verlag: Birkhäuser
Erscheinungsdatum: 2011
Einband: Softcover
Zustand: New