Description: The Asymptotic Behaviour of Semigroups of Linear Operators by Jan van Neerven 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, Weiss 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. FORMAT Paperback LANGUAGE English CONDITION Brand New Publisher Description 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 Gelfands 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 Lyapunovs 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, Lyapunovs 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. Notes Springer Book Archives Table of Contents 1. Spectral bound and growth bound.- 1.1. C0—semigroups and the abstract Cauchy problem.- 1.2. The spectral bound and growth bound of a semigroup.- 1.3. The Laplace transform and its complex inversion.- 1.4. Positive semigroups.- Notes.- 2. Spectral mapping theorems.- 2.1. The spectral mapping theorem for the point spectrum.- 2.2. The spectral mapping theorems of Greiner and Gearhart.- 2.3. Eventually uniformly continuous semigroups.- 2.4. Groups of non-quasianalytic growth.- 2.5. Latushkin - Montgomery-Smith theory.- Notes.- 3. Uniform exponential stability.- 3.1. The theorem of Datko and Pazy.- 3.2. The theorem of Rolewicz.- 3.3. Characterization by convolutions.- 3.4. Characterization by almost periodic functions.- 3.5. Positive semigroups on Lp-spaces.- 3.6. The essential spectrum.- Notes Ill.- 4. Boundedness of the resolvent.- 4.1. The convexity theorem of Weis and Wrobel.- 4.2. Stability and boundedness of the resolvent.- 4.3. Individual stability in B-convex Banach spaces.- 4.4. Individual stability in spaces with the analytic RNP.- 4.5. Individual stability in arbitrary Banach spaces.- 4.6. Scalarly integrable semigroups.- Notes.- 5. Countability of the unitary spectrum.- 5.1. The stability theorem of Arendt, Batty, Lyubich, and V?.- 5.2. The Katznelson-Tzafriri theorem.- 5.3. The unbounded case.- 5.4. Sets of spectral synthesis.- 5.5. A quantitative stability theorem.- 5.6. A Tauberian theorem for the Laplace transform.- 5.7. The splitting theorem of Glicksberg and DeLeeuw.- Notes.- Append.- Al. Fractional powers.- A2. Interpolation theory.- A3. Banach lattices.- A4. Banach function spaces.- References.- Symbols. Promotional Springer Book Archives Long Description 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 Gelfands 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 Lyapunovs 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, Lyapunovs theorem implies that the expo Details ISBN3034899440 Short Title ASYMPTOTIC BEHAVIOUR OF SEMIGR Series Operator Theory: Advances and Applications Language English ISBN-10 3034899440 ISBN-13 9783034899444 Media Book Format Paperback Series Number 88 Birth 1964 Edition 96199th Year 2011 Publication Date 2011-10-01 Country of Publication Switzerland Illustrations XII, 241 p. Imprint Birkhauser Verlag AG Place of Publication Basel Author Jan van Neerven Pages 241 Publisher Birkhauser Verlag AG Edition Description Softcover reprint of the original 1st ed. 1996 Alternative 9783764354558 DEWEY 510 Audience Professional & Vocational We've got this At The Nile, if you're looking for it, we've got it. With fast shipping, low prices, friendly service and well over a million items - you're bound to find what you want, at a price you'll love! TheNile_Item_ID:96317745;
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ISBN-13: 9783034899444
Book Title: The Asymptotic Behaviour of Semigroups of Linear Operators
Number of Pages: 241 Pages
Language: English
Publication Name: The Asymptotic Behaviour of Semigroups of Linear Operators
Publisher: Springer Basel
Publication Year: 2011
Subject: Mathematics
Item Height: 235 mm
Item Weight: 394 g
Type: Textbook
Author: Jan Van Neerven
Item Width: 155 mm
Format: Paperback
