A-XY~*的Moore-Penrose逆

设A是一个C*-代数,对于任意的HilbertA-模K和H,令L(H,K)表示K到H上的可共轭算子全体,A是L(H,H)的一个可逆元,X,y是L(K,H)上的两个算子且满足X,Y,A-XT*都有闭值域.记X1=A-1X,Y1=(A-1)·Y,QX1=IH-X1X+1,QY1=IH-Y1Y+1,其中IH是H上的恒等算子,X+1,Y+1分别是X,Y的Moore-Pence逆.证明了Moore-Penrose逆(A-XY*)*=QX1A-1QY1的充分必要条件是:Y*1XY*1=Y*1,且XY*1X=X.

The Moore-Penrose inverse of A-XY~*

Let A be a C*-algebra.For any Hilbert A-modules K and H,let L(K,H)denote the set of the adjointable operators from K to H.Let A be an inverse element of L(H,H),and X,Y be two elements of L(K,H) such that X,Y and A-XY* have closed ranges.Let X1=A-1X,Y1=(A-1)*Y, QX1=IH-X1X+1 and QY1=IH-Y1Y+1,where IH is the identity operator on H,and X+1,Y+1 are the Moore-Penrose inverses of X1 and Y1respectively.In this note,we prove that the Moore-Penrose inverse(A-XY*)+ equals QX1A-1QY1 if and only if Y*1XY*1=Y*1 and XY*1X...