正交投影的积与差的Drazin逆

Drazin inverses of products and differences of orthogonal projections

目的设P和Q是B(H)中的两个正交投影,利用P与Q的算子矩阵的形式,给出正交投影P和Q的积与差的Drazin可逆性的等价刻画。方法利用算子矩阵的分块技巧,根据Drazin可逆性的定义及其相关性质推导。结果得出PQ(resp.P-Q)是Drazin可逆的充要条件是Q0(resp.I-Q0)是可逆的。同时,给出正交投影的积PQ和差P-Q的Drazin逆的表达式。结论得出两正交投影的积与差的Moore-penrose可逆性和Drazin可逆性是一致的。

Let P and Q be two orthogonal projections on a Hilbert space with the operator matrix forms. Aim To discuss the Drazin invertibility of products and differences of orthogonal projections on a Hilbert space, and give the necessary and sufficient conditions for PQ （respectively, P-Q） has the Dazin inverse. Methods The technique of block operator matrices is used to investigate the Drazin invertibility of products and differences of orthogonal projections. Results It is get that PQ （respectively, P-Q） is Drazin invertible if and only if Q0 （respectively, I- Q0 ） is invertible. Meanwhile, the Drazin inverses of PQ and P--Q are estabilished. Conclusion PQ （respectively, P-Q） has the Dazin inverse if and only if PQ （respectively, P-Q） has the Moore-Penrose inverse.