{"product_id":"the-asymptotic-behaviour-of-semigroups-of-linear-operators-von-jan-van-neerven-und-jan-van-neerven","title":"The Asymptotic Behaviour of Semigroups of Linear Operators","description":"\u003cp\u003eOver 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.\u003c\/p\u003e\u003cdiv class=\"aw-variant-hidden-subtitle-div\" id=\"aw-variant-subtitle-9783034899444\"\u003e\u003ch3\u003e\u003c\/h3\u003e\u003c\/div\u003e\u003cdiv class=\"aw-variant-hidden-subtitle-div\" id=\"aw-variant-subtitle-9783764354558\"\u003e\u003ch3\u003e\u003c\/h3\u003e\u003c\/div\u003e","brand":"Autorenwelt Shop","offers":[{"title":"Softcover - 9783034899444","offer_id":49592316723525,"sku":"9783034899444","price":106.99,"currency_code":"EUR","in_stock":true},{"title":"Hardcover - 9783764354558","offer_id":49593035686213,"sku":"9783764354558","price":106.99,"currency_code":"EUR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0940\/0622\/files\/331ca6d8-747f-448c-b734-b7feaad91f14.jpg?v=1772173658","url":"https:\/\/shop.autorenwelt.de\/products\/the-asymptotic-behaviour-of-semigroups-of-linear-operators-von-jan-van-neerven-und-jan-van-neerven","provider":"Autorenwelt Shop","version":"1.0","type":"link"}