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By V.P. Maslov

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

Example text

Thus U(t) - exp{itA}, A E G(M,O). To prove (3), we note that for arbitrary x E X, Ra(An)x - Ra(A)x . i(f C0+f C if a- ixt Ce it Anx_Q((t)xldt = )e-tatEelt 0x - U(t)xldt E II +1 21 Im). < 0, C > 0 (see (4), item A). If A lies in a compact subset of the lower half-plane, Im. G -6 < 0. It follows that, given c > 0, we can find such C that III2211 < < E/2 for all n (the norm of the integrand does not exceed 2M 11xll exp(-it)). )x converges to U(t)x uniformly for t C CO, C], so ,Illi< E/2 for n large enough, and (3) is therefore proved.

The generator o? the semigroup (1) is the operator A = -i(8/3x) with the domain DA, consisting of all absolutely continuous functions f e L2(R1) such that f/2x E L2(tI). The operator S so defined is self-adjoint. Indeed, C'0(al) is invariant under the action of the semigroup (1), and ( i dt 1U(t)f(x)1)1t=0 = -i 3ff(x) , f GCo(Itl), (2) so that the restriction of A on Co(htl) is -i(a/ax). It follows from Theorem 3 of the preceding item, that A is the closure in LZ((tl) of -i(2/Bx), defined on Co(al).

0 (2) as n w m for some family U(t) of operators in X, it is necessary that s R1(An) : R1(A), IrnA < -w (3) 47 for some densely defined operator A in X, and sufficient that* Rao(An) I Rao(A) (4) for some A , Im)< -a. In this case Ac- G(M,w) (in particular, A is then closed) 5J U(t)o- exp{itA}. Proof. We may assume that w - 0, considering An + iw instead of A. If (2) holds, d(t) is a strongly continuous semigroup, (a) Necessity. satisfying AU(t)l G M. Indeed, continuity and the estimate for the norm immediately follow from (2); the semigroup property is proved in complete analogy with that in Theorem 1, (f) of item A.

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