Moore-Penrose逆的可微性

Differentiability of Moore-Penrose Inverse

讨论了Hibert空间H1到H2有界线性算子全体构成的Banach空间LH1,H2上Moore-Penrose逆的连续性和可微性,给出了函数T+t在一点可微的几个等价描述,同时得到一个求导公式.所得的结果推广了Golub和Pereyra早期的主要结果.

Let H1, H2 be two Hiberts spaces over the complex field, and let L（H1, H2） denote the Banach space of all bounded linear operations T： H1→ H2 with the operator norm ｜｜T｜｜ = sup { ｜｜ Tx｜｜：｜｜x｜｜ = 1 }. Let [a, b] be an interval with be an element of [a, b], and T（t） be an operator valued function defined for all t ∈ [a, b]. By T＇（t） denote the derivative of T（t） at t, and by T＇（t） denote the Moore-penrose inverse T（t）^＋ of T（t）. The continuity and differentiability of the Moore-penrose inverse in L （H1, H2） is investigated. Some necessary and sufficient conditions for the function is differentiable at to are given. A formula for derivative （T^＋） ＇ is derived. The main results of Golub and Pereyra in [4] generalized to the case of operators in Hibert spaces.