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 \(A\in B(H)\) and \(B\in B(K)\). We denote by \(M_C\) the operator acting on \(H\oplus K\) of the form \(M_C=\left( \begin{array}{cc}A&{}C\\ 0&{}B\\ \end{array}\right) \). In this paper, we give necessary and sufficient conditions for \(M_C\) to be an upper semi-Fredholm operator with \(n(M_C)>0\) and \(\hbox {ind}(M_C)<0\) for some left invertible operator \(C\in B(K,H)\). Meanwhile, we discover the relationship between \(n(M_C)\) and n(A) during the exploration. And we also describe all left invertible operators \(C\in B(K,H)\) such that \(M_C\) is an upper semi-Fredholm operator with \(n(M_C)>0\) and \(\hbox {ind}(M_C)<0\).