(O,A)类算子半群序列的收敛性

ON THE CONVERGENCE OF A SEQUENCE OF(O,A)SEMIGOUPS OF LINEAR OPERATORS

设 X 为 Banach 空间,{T_n(t))是 X 上的(o,A)类算子半群序列。文献1,2和3的作者分别讨论了(Co)类、(1,A)类和(A)算子半群序列的收敛性,在这篇文章中我们证明了:若 T_n(t),T(t)∈(0,A),并满足条件:(1)T(t)与 T_n(s)可交换(n=1,2,…,t,s>0),(2)对任-t>0和 x∈X,sup■T_n(t)x■<+∞,且存在实数 w 和 M,>0使∫_0^(+∞)e^(-wt)sup■T_n(t)x■dt≤M_x和∫_0^(+∞)e^(-wt)■T(t)x■dt≤M_x,则 s—■T_N(t)=T(t)当且仅当 s-■R(λ;A_n)=R(λ;A)(Reλ>w),并且我们也建立了(0,A)类半群序列的一个收敛定理,所得结果推广了文献1,2和3的若干结论。

Let X be a Banach space and{T_n(t)} be a sequnce of(0,A)-semigroups on X.Authors of References 1,2 and 3 discuss the convergence of a sequence of (C_0),(1,A)and(A)-semigroups,respectively.In this paper we prove that if T_n(t),T(t)∈(0,A)and satisfy following conditions:(1)T(t)commutes with T_n(s)(n=1,2,…;s>0);(2)For any t>0 and x∈X Sup‖T_n(t)x‖ <+∞,and there exist real numbers ω and M_s>0 such that ∫_0^(+∞)exp(-ωt)sup‖T_n(t)x‖dt≤m_s, and ∫_0^(+∞)exp(-ωt)‖T(t)x‖dt≤M_s, then s-■ T_n(t)=T(t)if and only if s-■ R(λ;A_n)=R(λ;A),where Reλ >ω.Moreover,We establish also a theorem on convergence of a sequence of(0, A)-semigroups.Obtained results extend some results of References 1,2 and 3