Fredholm complements of upper triangular operator matrices

For a given operator pair (A,B)∈(B(H),B(K)), we denote by M_C the operator acting on a complex infinite dimensional separable Hilbert space H⊕K of the form M_C=(AC0B). This paper focuses on the Fredholm complement problems of M_C. Namely, via the operator pair (A, B), we look for an operator C∈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 2×2 upper triangular operator matrices is investigated by the virtue of the so-called entanglement spectra of the operator pair (A, B).