By Krein S., Khazan

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Extra info for Semigroups of operators, cosine operator functions, and linear differential equations

Example text

In particular, ,h~(i,ul--A) = {0} if and only if ,4,~ -- {0} . 10. , is bounded. 11. p. p. group. 12. [144] A group exp(tA) is uniformly almost periodic if and only if the following three conditions are satisfied: (i) the group exp(tA) is uniformly bounded; (ii) the set (1/i)e(A) is a harmonious subset of R; (iii) the set of linear combinations of the characteristic vectors of the generating operator A is dense in the space E. 13. [144] If exp(tA) is uniformly almost periodic, then e(A) consists of simple poles of the resolvents R(A,A) and a(A) = Pa(A).

P. group. 6. [144] Assume that the group exp(tA) is almost periodic. Let it/(r/6 R) be an isolated point of the spectrum ~r(A). 7. /F(igl--A), such that (x,f) ~ 0. p. 8. [15] Let A6~,(M, 0) and let the space E be reflexive. p. 9. p. then nul (it,d - - A ) = nul (igl*--A *) 1087 for all # E R. In particular, ,h~(i,ul--A) = {0} if and only if ,4,~ -- {0} . 10. , is bounded. 11. p. p. group. 12. [144] A group exp(tA) is uniformly almost periodic if and only if the following three conditions are satisfied: (i) the group exp(tA) is uniformly bounded; (ii) the set (1/i)e(A) is a harmonious subset of R; (iii) the set of linear combinations of the characteristic vectors of the generating operator A is dense in the space E.

4. [22] For any x ~ E the function ~ satisfies Eq. 1) there exists an x ~ E such that v(t) -~-~Zl,(t)x. 5. [22] Let E be adjoint to some Banach space F and for a particular t o > 0 assume the operator is adjoint to some operator that is bounded in F. Then x ~ E belongs to E if and only if sup 11o21(t)x II ~ o~. 6. [22]. If E is reflexive, then x ~ E_ belongs to E if and only if the function I~ [] is bounded close to zero. For the case of analytic semigroups it has been possible to establish a connection between the behavior of the function ~ ( t ) x and membership of bounded values of x in certain spaces intermediate between E and E_.