By Francis J Murray

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Thus for A >0, hypothesis, R(A I-A) = X, 0 (A I-A) is onto. 0 and belongs to iS(X) and therefore R(A^,A) operator suppose A is x^— 0 and X, (A I-A)x - ~ n Indeed, closed, y -^y A x-y € X and 0 let in X. hence Hence We x (A I-A) n show R(A ,A) exists 0 is closed. (x ,y ) € n (AI-A) is 1-1 and since, by r(A), that 0 This implies that the the graph (x,y) R(A^,A)(A I-A)x - 0 n 0 n € of F(A). A, and Clearly R(A^,A)(A^x-y). Thus we have x = R(A^, A) (A^x-y). Since R(A^,A)X c D(A), we have x € D(A) and hence (A^I-A)x = (A^x-y) implying thereby proving that A is closed.

13) where r stands for the gamma function. 14) T(t)x dt. 5. j. for all n € N q , and hence we have ^ llalli' all X >0. j. ^ M llxllco and hence X = X,j, ^ Xoo and, therefore, one can freely exchange these spaces without destroying topological properties. 4. Let ADJOINT SEMIGROUPS. A be Banach the space operators infinitesimal X and -{T*(t), ta:0 in the dual X*. D(A ) is dense in X generator the of a C^-semigroup corresponding In the case of a general nor t— > T (t) -{T(t), t a s O i n adjoint semigroup Banach space, is strongly continuous a of neither (in the norm topology of X*) on [0,oo).

For A,fi>0, for all ? 5) and § € X, we have (2. 7) R(A,A)-R(fx,A) = (fx-A)R(A,A)R(p,A) = (p-A)R(p, A)R(A, A) for all A,/Ji € p(A). R(A,A) and R(/i,A) This expression is called the resolvent identity. commute and hence one can easily verify that all Thus the operators A ,A ,T. (t),T (t) commute for all A,jn € p(A) and t^O. j do. 26 (2. 2. 8) for all tao. 9) for all A > 0 and tao. I Now we are prepared to prove the famous Hille-Yosida theorem which forms the corner stone of semigroup theory. 8. (Hille-Yosida) Let X be a Banach space and A € ub (X) with D(A) and R(A) in X.