希尔伯特空间上的Trotter乘积公式与线性发展方程--纪念加藤寿一百岁
Trotter product formula and linear evolution equations on Hilbert spaces On the occasion of the 100th birthday of Tosio Kato
摘要：
本文研究了可分希尔伯特空间上的发展方程：$\partial$ $\partial$ t u（t）= -（A + B（t））u（t），t $\in$ I = [0，T ]，其中A是非负自伴算子，B（$\times$）是一类非负自伴算子，使得dom（A $α$）$\subseteq$ dom（B（t））对于某个$α$ $\in$ [0，1），并且映射A-$α$ B（$\times$）A-$α$是H{<$}lder连续的，其H{<$}lder指数为$β$ $\in$（0，1）。证明了在满足$β$ > 2 $α$- 1的条件下，发展方程的解算子U（t，s）可以用A和B（t）生成的半群的组合在算子范数下逼近.近似的收敛速度由H{<$}lder指数$β$给出。利用演化半群方法证明了这一结果。
英文摘要：
The paper is devoted to evolution equations of the form $\partial$ $\partial$t u(t) = --(A + B(t))u(t), t $\in$ I = [0, T ], on separable Hilbert spaces where A is a non-negative self-adjoint operator and B($\times$) is family of non-negative self-adjoint operators such that dom(A $α$) $\subseteq$ dom(B(t)) for some $α$ $\in$ [0, 1) and the map A --$α$ B($\times$)A --$α$ is H{ö}lder continuous with the H{ö}lder exponent $β$ $\in$ (0, 1). It is shown that the solution operator U(t, s) of the evolution equation can be approximated in the operator norm by a combination of semigroups generated by A and B(t) provided the condition $β$ > 2$α$ -- 1 is satisfied. The convergence rate for the approximation is given by the H{ö}lder exponent $β$. The result is proved using the evolution semigroup approach.
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