By Charles J. K. Batty, Derek W. Robinson

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**Example text**

Thus for a s D ( H ) ~o(Ha) = lim 0~((I - St)a)/t >10. , H is p-dissipative. , S is p-contractive. In particular a Co-semigrou p is a contraction semigroup if, and only if, its generator is norm-dissipative. This is in fact part of the statement of the Hille-Yosida theorem because norm-dissipativity corresponds to the bounds It

This fact makes the equivalence of Conditions 2 and 3 in the following theorem a little less surprising. 1. Let H; D(H)~--,~ be a norm densely-defined linear operator on the real Banach space ~ and p a half-norm on ~ . The following conditions are equivalent: 1. p((l + otn)a) >i p(a) for all (small) ~ > O, and all aeD(H), 2. to(Ha) >10 for some toedp(a), and for all aeD(H), 3. to(Ha) >10 for all toedp(a), and for all aeD(H). Proof. 1 r This equivalence follows from the above relation (*) with the choice b = Ha.

But I1"I1~is equivalent to II il, because 2 + is normal, and hence S is a Co-semigrou p on (2, I1"11) with IIs, II ~ M exp{rt} for some M/> 1. Finally, positivity of the semigroup follows from positivity of the resolvents (I + ~tH)- 1. [] The next corollary gives a variant of this result which stresses the N -dissipativity. 8. Assume 2 + is normal and int 2 + ~ 0. Let N u and ,~, denote the half-norms associated with a u e int ~' +. Then the following conditions are equivalent: 1. H generates a positive Co-semigrou p S with S,u <~u (respectively S,u = u) for all t > O, 2.