Fredholm complements of upper triangular operator matrices

For a given operator pair \((A,B)\in (B(H),B(K))\) $(A,B)\in (B(H),B(K))$ , we denote by \(M_C\) $M_C$ the operator acting on a complex infinite dimensional separable Hilbert space \(H\oplus K\) $H\oplus K$ of the form \(M_C=\bigl ( {\begin{matrix} A&{}C\\ 0&{}B \\ \end{matrix}}\bigr )\) $M_C=(\begin{array}{cc}A& C\\ 0& B\\ \end{array})$ . This paper focuses on the Fredholm complement problems of \(M_C\) $M_C$ . Namely, via the operator pair (A, B), we look for an operator \(C\in B(K,H)\) $C\in B(K,H)$ such that \(M_C\) $M_C$ is Fredholm of finite ascent with nonzero nullity. As an application, we initiate the concept of the property (C) as a variant of Weyl’s theorem. At last, the stability of property (C) for \(2\times 2\) $2\times 2$ upper triangular operator matrices is investigated by the virtue of the so-called entanglement spectra of the operator pair (A, B).