The Upper Semi-Weylness and Positive Nullity for Operator Matrices

Let H and K be infinite dimensional separable complex Hilbert spaces and B(K,H) the algebra of all bounded linear operators from K into H.Let A is an element of B(H) and B is an element of B (K). We denote byMCthe operator acting on H circle plus K of the form MC=(AC0B).In this paper, we give necessary and sufficient conditions for MC to be an upper semi-Fredholm operator with n(MC)>0 and ind(MC)<0 for some left invertible operator C is an element of B(K,H). Meanwhile, we discover the relationship between n(MC) and n(A) during the exploration. And we also describe all left invertible operator sC is an element of B(K,H)such that MC is an upper semi-Fredholm operator with n(MC)>0 and ind (MC)<0.