Discussion on Matrices Fixed Nullity in Complement Problem of Operator Matrices

Let H and K be separable infinite-dimensional Hilbert spaces, and let A is an element of B (H) and B is an element of B(K) be given operators. We denote by MC the operator acting on H circle plus K of the form M-C = (AC(0B)). In this paper, some necessary and sufficient conditions are obtained for M-C to be a Fredholm operator with n(M-C)>0 and ind(M-C)<0 for some left invertible or invertible operator C is an element of B (K,H). Meanwhile, for the nullity of M-C, we discuss the relationship between n(M-C) and n(A) by different method. As the application of above results, the weak properties of Weyl's theorem for upper triangular operator matrices are explored.