Positive one-parameter semigroups on ordered banach spaces by Charles J. K. Batty, Derek W. Robinson

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

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Fill)-1(/. + o~H)-na, ]l + + 11(/. + f i n ) - 1 ( 1 + o:H)-"a 2 ]]} ~/r{ II(I + f i B ) - ' a, II § II(I + f l U ) - ' a s It }. < [l(/+ ~n)na 11'. It now follows by a standard perturbation argument and the F e l l e r - M i y a d e r a Phillips theorem that H generates a Co-semigrou p S on ~ with the bounds IIS, II' "< ~ e'/a and hence [1St II ~ Me'/P for some M / > 1. 12. [] We conclude this section with two examples. The first is a c o m m e n t on dissipativity in an algebraic setting. 16 (Dissipations on C*-algebras).

2 the assumption of positive attainment is a somewhat implicit condition on M. 8 to obtain the following statement. 1 (resp. 2) are equivalent. The first option of this corollary, the Riesz norm, covers Banach lattices, C*- POSITIVE ONE-PARAMETER SEMIGROUPS ON ORDERED BANACH SPACES 263 algebras, and their duals. The second option is equivalent to the norm on ~ being an order-norm and the dual-norm being monotone. 1. 4. Let ( ~ , ~ + , IIl]) be an ordered Banach space for which ~+ is approximately dominating and IIlf is monotone.

Me~"-~' ifs<~t. Hence IIu + ~H)-la H= f o ~ dt e - tS ta fs/a f o s/~ <~MS/~ + eM e-S/U(1 - ~7). But since ~ and s are arbitrarily small, this shows that Condition 1 is false. 2=, 1. Suppose ~ + has a base, so that there is a fl > 0 such that fl i=~--1a i ~ i=1~'~ 11ai 11 for all n i> 1 and all a i t ~ + . Hence for a e ~ + one can use a Riemann approximation to establish that II(I + ~ H ) - l a tl = fo % >1f l - ' dt e - ' S , a dt e-t~o(Tt)t1 a [1. POSITIVE ONE-PARAMETER SEMIGROUPS ON ORDERED BANACH SPACES But Condition 2 implies that tp(s) > 0 for all s and hence Condition 1 is valid.

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