An Introduction to Functional Analysis (North-Holland Texts by Mischa Cotlar

By Mischa Cotlar

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K IC Proof. l eML ;)f - L"f II = e " Y 11 n kt k=o (Lk - L')f enkt IILkf - L"f II. But 0. IILkf-L"fIl5IILik-"if-flland Lm-I= (L`-L-- t), whence IILmf-fII5 i=t IILf-L'-tfII. IlLf--f11. ( p nk 1/2 {kYx. nk 1;2 (k - t)2 IILf - f II o k? lcomputation (cf. any elementary by the Schwarz inequality. But an easy probability text where the mean and variance of the Poisson distribution are Se "{ computed) shows that x nk so k! and the lemma follows. (k -- n)2 = re" Some Applications 8.

By hypothesis on R = R(Ao), A' _ {0} and . is dense. Define Ao : 9 (AD) = . -+ f by AD = Al o- R(A°)-- i Clearly (AD - Ao)R(Ao) = R(Ao)(Ao - AD) = I . on -4. Moreover, for A e E one has the following calculation for operators defined on X: (A - Ao)R(A) = [(A - Ao) + (A° Ao)]R(A) = [(A - AD) + (AD - Ao)]R(Ao)[I - (A - A0)R(A)] = I + (A - Ao)[R(Ao) - R(A) - (A - A0)R(A)R(Ao)] = I and similarly R(A)(A - AD) = I on 9t. Consequently R(A) = (A - AD)-' for all A E E. The estimate II(A - A°)-"f II = m)imz II(A - Am)-"III 5 MIII II Re(A)° shows that A° generates a (CO) semigroup.

1 is reflexive. (f) X, X can happen if X is not reflexive. Proof. 1 for each at > 0. Let f e X and let g. = (I - aA)- if e 2(A) c 37o. } be a sequence of positive numbers converging to zero. A)-'f II S (If ll for each n. If X is re- - flexive, then by the weak sequential compactness of closed balls in X, has a weakly convergent subsequence, which we take to be the whole sequence {g,,} (by redefining the Then go - weak lim, g belongs to the weak closure of 2(A), which is Xo. ) = go - f. Thus weak lim....

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