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

Neerven, Jan Van

 
9783034899444: The Asymptotic Behaviour of Semigroups of Linear Operators: 88 (Operator Theory: Advances and Applications)
Tapa blanda
ISBN 10: 3034899440 ISBN 13: 9783034899444
Editorial: Birkhäuser, 2011

Sinopsis

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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Reseña del editor

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.

Reseña del editor

This book presents a systematic account of the theory of asymptotic behaviour of semigroups of linear operators acting in a Banach space. The focus is on the relationship between asymptotic behaviour of the semigroup and spectral properties of its infinitesimal generator. The most recent developments in the field are included, such as the Arendt-Batty-Lyubich-Vu theorem, the spectral mapp- ing theorem of Latushkin and Montgomery-Smith, Weis's theorem on stability of positive semigroup in Lp-spaces, the stability theorem for semigroups whose resolvent is bounded in a half-plane, and a systematic theory of individual stability. Addressed to researchers and graduate students with interest in the fields of operator semigroups and evolution equations, this book is self-contained and provides complete proofs.

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Editorial
Birkhäuser
Año de publicación
2011
Idioma
Inglés
ISBN 10 
3034899440
ISBN 13 
9783034899444
Encuadernación
Tapa blanda
Número de páginas
256
Contacto del fabricante
no disponible
Persona responsable
no disponible

Otras ediciones populares con el mismo título

9783764354558: The Asymptotic Behaviour of Semigroups of Linear Operators: 88 (Operator Theory: Advances and Applications)

Edición Destacada

ISBN 10:  3764354550 ISBN 13:  9783764354558
Editorial: Birkhäuser, 1996
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