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Extra info for Complex Analysis and Dynamical Systems III

Example text

5). We set S S - T ~ ~ T whenever O. As usual ( is ordered by R+. 2 Examples. 11). b) Every complexification of a real order unit space (see , V). c) Every C* -algebra A. The real space Ao consists of the selfadjoint elements, A+ consists of the nonnegative selfadjoint elements. The most important result of  now is the following one: Let t= (T ) ~ denote a strongly continuous semigroup of positive linear opert t::::o ators on the ordered Banach space E( over (. Let A be the infinitesimal generator of t and denote by o(A) its spectrum and by seA) the spectral bound.

Are •• defined in the sections just mentioned. "H ". "A-regular open set" are as defined in (3) i "A-Cauchy regular open set" has the meaning defined in 15) ; "Dirichlet-regular open set". 3. the ten. inside II lIe(K) are to be understood as being the corresponding restrictions to K. Lumer 68 open set", has the meaning defined in [23). Throughout this section, (22) A will be a local operator on n, real, locally dissipative. 3 - consider our local operator A as restricted to the corresponding real functions.

Oj separates the points £E. the dual space E' then .. is strongly asymptotically stable. PROOF. Since Im(A) + separates the points of E', the linear hull X of Im(A)+ is dense in E. If x E Im(A)+ then the weak limit z := o~ Tsxds exists by Prop. 2 because of our assumption. 3) this implies that ( (t T xds) oJ s assertion now follows from Prop. 2. t>o converges strongly to z. 3 The Main Result. If a strongly continuous semigroup = (T t )t"'o , is exponentially asymptotically stable on the whole space then by the uniform boundedness principle +is uniformly exponentially asymptotically stable hence its type Wo (see Sect.