The Invertible Completions And Fredholm Completions Of Partial Operator Matrices

Let H and K be infinite dimensional Hilbert spaces.First,for given operators A?B(H),B E B(K)and C?B(K,H),under the condition that R(B)is closed,the equivalent condition is given for to be left(right)invertible for all X?B(H,K).Next,for given operators A?B(H),B?B(K,H)C?B(K)and D?B(K),we determine the necessary and sufficient condition so that there exist X? B(H,K),Y?B(H),Z?B(K,H)and W?B(H,K)such that is a Fredhom operator with Finally,some examples are given to illustrate the effectiveness of our results.