Continuous semigroups on ordered Banach spaces (Q791123)

The Hille-Yosida theorem establishes that a linear operator H generates a \(C_ 0\)-contraction semigroup if and only if H is norm closed norm densely defined and satisfies both the range condition \(R(I+\alpha H)=\beta\) and the dissipativity condition, \(\|(I+\alpha H)a\| \geq \| a\|,a\in D(H)for\alpha>0.\) The author obtains a direct analogue for \(C_ 0\)-semigroup S on an ordered Banach space \((\beta,\beta_+,\|.\|)\) with the additional property \(S\beta_+\subseteq \beta_+\) in the following theorem. Let H be a linear operator on the ordered Banach space \((\beta,\beta_+,\|.\|)\) and let N denote the canonical half-norm associated with \(\beta_+\). Assume \(\|.\|\) is a Riesz norm. The following conditions are equivalent. (1) H generates a \(C_ 0\)-semigroup S of positive contractions (2) H is norm-densely defined, norm-closed, and satisfies both the range condition \(R(I+\alpha H)=\beta\) and the dissipativity condition \(N(I+\alpha H)a\geq N(a),a\in D(H),\) for all small \(\alpha>0.\)