Riesz函数演算的Lipschitz性质

Lipschitz Properties of Riesz Functional Calculus

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)．

Let A be a unital complex Banach algebra, Ω∪→C be a region and γ∪→C be a rectifiable closed curve such that ins（γ）∪→Ω . It is proved that the Pdesz functional calculus f： x → f（x） is a Lipschitz operator from some Aδ^γ into A, i.e., f ∈ L^1 （Aδ^γ, A） and has the Lipschitz constant L1（f）≤Mf（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.