Fredholm complements of upper triangular operator matrices

Abstract

For a given operator pair \((A,B)\in (B(H),B(K))\), we denote by \(M_C\) the operator acting on a complex infinite dimensional separable Hilbert space \(H\oplus K\) of the form \(M_C=\bigl ( {\begin{matrix} A&{}C\\ 0&{}B \\ \end{matrix}}\bigr )\). This paper focuses on the Fredholm complement problems of \(M_C\). Namely, via the operator pair (A, B), we look for an operator \(C\in B(K,H)\) such that \(M_C\) 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 \(2\times 2\) upper triangular operator matrices is investigated by the virtue of the so-called entanglement spectra of the operator pair (A, B).