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**Additional info for Asymptotics of Operator and Pseudo-Differential Equations (Monographs in Contemporary Mathematics)**

**Sample text**

13) k-0 k Hence we. obtain, using the inequality Itk - TkI G It - exp(itA)-exp(iTA)I < Ckll k ITI)k-1: (ItI+ITI)k+LAI Since the sum in the parentheses is finite, we conclude that U(t) is uniformly continuous. Vice versa, let U(t) be uniformly continuous. We assert that Lim -iuR(A) - I = 0. (14) 39 If it were already pruved, we may conclude that the operator -iuR_iU(A) has a bounded inverse for sufficiently large u and therefore A = -iU By (4) - (R-iu(A))-1 is bounded. The proof of (14) is as follows.

Tends to zero as u,v - 0D and we obtain that there exists the limit (16) U(t)x def Lim Uu(t)x, the convergence in (16) being locally uniform with respect to the variable The uniform boundedness of the family {Uu(t)} yields that the limit t. (16) exists for all x E X, U(t) is strongly continuous, IIU(t)II 6 :2im Uu(t)n 4 kim Me (tuw/u-w) Mew t Next we have to show that U(t) is a semigroup with the generator A. have, for an arbitrary x(=- X, We IU(t+r)x-U(t)U(r)XII +IIU(t)U(r)x-Uu(t)U(T)x +I]Uu(t)U(r)x-Uu(t)Uu(r)xI1, since the semigroup property is valid for UV(t).

Since r fe(w+Iml)ttm-1dt = (m-1)! (-Ima - w)m 0 , w+lma < 0, we obtain (5), and (e) is proved. Now we move on to the proof of uniqueness Let x - 0 and x(t) satisfy (3). We assert that x(t) vanishes in (d). identically. o prove it, consider an auxiliary X-valued function y(t,T) = U(t-T)X(r) defined for 0 < T G t. We have y(t, 0) = 0 and y(t, t) = x(t). It turns out that dy/dz exists and vanishes identically. This immediately yields x(t) __ y(t,t) - y(t,0) - 0. Let T E CO,t] be fixed, E E C-T,t- TI (so that T + E G 10,t]).