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Asymptotic Behaviour of Semigroups of Linear Operators

Asymptotic Behaviour of Semigroups of Linear Operators

          
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About the Book

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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Product Details
  • ISBN-13: 9783764354558
  • Publisher: Birkhauser Verlag AG
  • Binding: Hardback
  • Height: 234 mm
  • No of Pages: 241
  • Series Title: Operator Theory: Advances and Applications
  • Weight: 544 gr
  • ISBN-10: 3764354550
  • Publisher Date: 30 Jul 1996
  • Edition: 1996 ed.
  • Language: English
  • Returnable: N
  • Spine Width: 16 mm
  • Width: 156 mm


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