Riesz函数演算的Lipschitz性质

设A是具有单位的复Banach代数,Ω为复平面C上的一个区域,γ是复平面上的任一可求长的封闭曲线且其内部区域ins(γ)Ω,证明了存在A的子集A_δ~γ,使得对于Ω上的任一解析函数f,Riesz函数演算f:xf(x)是从A_δ~γ到A中的Lipschitz映射即f∈L~1(A_δ~γ,A)且其Lipschtiz常数(L_1(f)■M_f(γ)Γ)/(2πδ~2)．作为这一结果的应用,研究了算子值的根式函数TT~(1/m)及绝对值函数T|T|的Lipschitz性质．最后,证明了:若f为一个复值整函数,则对任一非空有界集EA,有f∈L~1(E,A)．

Lipschitz Properties of Riesz Functional Calculus

Let A be a unital complex Banach algebra,Ωbe a region andγbe a rectifiable closed curve such that ins(γ)Ω.It is proved that the Riesz functional calculus f:x)f(x) is a Lipschitz operator from some A_δ~γinto A,i.e.,f∈L~1(A_δ~γ,A) and has the Lipschitz constant L_1 (f)■(M_f(γ)Γ)/(2πδ~2)As an application,Lipschitz propertites of the operator valued root-function TT 1/m and the absolute value function T|T| are disccussed.Lastly,it is proved that f∈L~1 (E,A) holds for every nonempty bounded subset E of A.