By Gan Sh.

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An OS exp(tA) is said to be c o m p a c t beginning (or to b e c o m e c o m p a c t ) at t o if, for t > t 0, the operator exp(tA) is compact. If t o = 0 then the OS exp(tA) is said to be compact. 1. [126] Assume that an OS exp(tA) becomes compact. 2. [126] If the generating operator o f an OS exp(tA) has a c o m p a c t resolvent, and the OS itself is continuous with respect to norm at the point to, then the OS exp(tA) is c o m p a c t beginning at t o. THEOREM 1. [124] An OS exp(tA) is c o m p a c t if and only if it is continuous with respect to n o r m for any t > 0 and the generating operator has a c o m p a c t resolvent.

Applications are also discussed in these papers. Analytic OS's whose domains are not dense in E were considered in [113, 114, 187, 375]. Lipshits operator semigroups are discussed in [416, 417]. ) of mappings of the spectra of operator semigroups are considered in [126, 127]. 32, 4. 4. 35. 13. 30. 1073 3. 4). Recall that the generating operator (more accurately, the second generating operator) of a COF C(t,A) is defined as the limit Ax:--lim 2 (C(h, A)x--x) h-+O on all x E E for which it exists.

7] Let A 6 ~ ( t ~ , 07) and assume that a(--A) has no limit points in R+. Then (i) the linear hull of characteristic and root vectors of A is dense in E if there exists a function X(t) such that llC(t,A)ll~x(t) aria x ( t ) ~ k ( l q - l t l ) ' for t6R, ~I~0; (ii) when condition lim X(t)/t = 0 is satisfied, the COF C(t,A) is periodic with period 1 if and only if a(A) C_ {--(2rk) 2, k E N). 2. Supplementary Comments. 21 is also true when A(t) : R --* B(E) with separable E (see [428]). 1090 6.

### Convergence of two-parameter banach space valued martingales and the radon-nikodym property of banach spaces by Gan Sh.

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