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    Alley-side Pickup: 10am - 7pm

    3038 Hennepin Ave Minneapolis, MN
    612-822-4611

Open Daily: 10am - 10pm | Alley-side Pickup: 10am - 7pm
3038 Hennepin Ave Minneapolis, MN
612-822-4611
The Asymptotic Behaviour of Semigroups of Linear Operators

The Asymptotic Behaviour of Semigroups of Linear Operators

Neerven, Jan Van

Paperback

Series: Operator Theory: Advances and Applications, Book 88

CalculusGeneral Mathematics

ISBN10: 3034899440
ISBN13: 9783034899444
Publisher: Springer Nature
Published: Oct 1 2011
Pages: 241
Weight: 0.81
Height: 0.54 Width: 6.14 Depth: 9.21
Language: English
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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