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.