The Asymptotic Behaviour of Semigroups of Linear OperatorsOver 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. |
Contents
I | xi |
II | xii |
III | 6 |
IV | 13 |
V | 17 |
VI | 21 |
VII | 23 |
IX | 24 |
XXV | 117 |
XXVI | 124 |
XXVII | 128 |
XXVIII | 133 |
XXIX | 141 |
XXX | 145 |
XXXII | 159 |
XXXIII | 164 |
Other editions - View all
The Asymptotic Behaviour of Semigroups of Linear Operators Jan van Neerven No preview available - 1996 |
The Asymptotic Behaviour of Semigroups of Linear Operators Jan van Neerven No preview available - 1996 |
Common terms and phrases
A)xo admits abstract Cauchy problem admits a holomorphic apply arbitrary assertions are equivalent assume assumption Banach function space Banach space bounded operator BUC(R+ choose closed subspace Co-group Co-semigroup compact convergence Corollary countable define denote dense eigenvalue eigenvector following assertions Fourier transform Fourier type function space Hence Hilbert space Hölder's inequality holomorphic extension implies infimum integral inverse isometric L¹(R L¹(R+ Laplace transform Lemma Let xo lim sup limt linear operator linear span LP R+ neighbourhood non-quasianalytic norm one vectors obtain op(A orbit positive Co-semigroup positive semigroups Proof Proposition prove R(ik R(wo rescaling resolvent identity result right half-plane scalarly Section semigroup sequence shows so(A spectral bound spectral mapping theorem spectral synthesis u(iy uniform boundedness theorem uniformly bounded uniformly bounded Co-semigroup uniformly continuous uniformly exponentially stable uniformly stable w₁ w₁(T wo(T X₁