向量值柯西问题的适定性

Maximal Regularity for Abstract Cauchy Problems

设A为复Banach空间X上的稠定闭算子.我们证明了若抽象柯西问题具有C-适定性,则{λ∈C:Reλ≥0） ρ（A）且存在C>0,使得任给λ∈C,Reλ≥0,都有（λ-A）-1≤C/1+ λ.因而A必为X上某一有界解析半群的无穷小生成元.对于定义在有限区间[0,T]上的抽象柯西问题,我们亦得到了类似的结果.

Let A be a linear closed densely defined operator on a complex Banach space X. We show that if the abstract Cauchy problemhas the C-maximal regularity, then {λ∈C: Reλ≥0} p(A) and there exists C > 0 such that for every λ∈C, Reλ≥0, we have ||(λ-A)-1||≤C/1+||λ||.A thus generates a bounded analytic semigroup in X. Similar results for the abstract Cauchy problem on a bounded interval [0, T] are also given.