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By Makhnev A. A.

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We define x(~) - ~(t)S(t)xdt, 47 The Infinitesimal Generator and we remark that lim ~(S(h) 1 - I)x(~) h$O = l i m e (f0 +~ ~ ( t ) S ( t + h)x d t - f0 +~ ~ ( t ) S ( t ) x dt ) = l h$O i m l-h (Jh +~ ~(t - h ) S ( t ) x at - fO+~ ~ ( t ) S ( t ) x dt ) . For s e (-cxz, a), we have ~(s)= 0, and consequently lim -~(S(h) 1 - I)x(~) h$O = l ih40 m e-h (f0 +~ ~(t - h ) S ( t ) x dt - ~ ( t ) S ( t ) x dt ) - fo +~ lim ~(t - h) - ~(t) S ( t ) x d t - - x ( ~ ' ) . 0 h Accordingly, x(9) e D ( A ) and A x ( ~ ) = -x(~').

7. Let ~t be a nonempty open and bounded subset in ~n whose boundary r is of class C 1. Then [[" ll: H I ( ~ ) --~ R+, defined by [[Ull_ (][~TUl[22(Vt)+IlU[FII~2(F))1/2 for each u E H I ( ~ ) , is a norm on HI(a) equivalent with the usual one. e. [[. [[o " H~(~) -+ ]~+, defined by II llo -IIWlIL ( ), for each u e Hl(~t), is a norm on H~(~) (called the gradient norm) equivalent with the usual one. In respect with this norm the application D" HI(~t) ~ H - I ( ~ ) , defined by (V, DU)H~(~),H-I(~ ) -- V u V v dw, 20 Preliminaries is the canonical isomorphism between H~(~t) and its dual H-I(f~).

3. Let A : D ( A ) C_ H --+ H be a linear operator. If A is densely defined, there exists its adjoint A* : D(A*) C_ H --+ H, and: (i) graph (A*) = {(x, f) C H xH; (f, y) = (x, g), V(y, g) C graph (A)} ; (iN) (x, f) e graph (A*) if and only if ( - f , x) E (graph (A))-L ; (iii) graph (A*) is closed in U x U. Preliminaries 24 P r o o f . Let G* - {(x, f) e H x H; (f, y} - (x, g), V(y, g) e graph (A)}. If (x, f) C G*, we have (f, y} - (x, Ay} for any y E D(A) and therefore I(x, Ay}[ < I]f[[llyl]. It follows that x e D(A*) and f - A ' x , and accordingly G* C_ graph (A*).

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3-Characterizations of finite groups by Makhnev A. A.


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