Some Fredholm left invertible completion problems for operator matrices

The structure of the weak approximate spectrum of two-by-two upper triangular operator matrix M_C = {pmatrix} A & C \\ 0 & B {pmatrix} acting on a Hilbert space H ⊕ K is studied. First, we characterize the relationship between the spectrum σ(M_C) and the weak approximate spectrum σFa(M_C) and give the equivalent conditions that make σFa(M_C) = σ(M_C) and σFa(M_C) = σFa(A) ∪ σFa(B) according to the properties that the operators A and B satisfy. Then, we study the weak property (ω_1) of M_C and explore the relationship between σ(M_C) = σFa(M_C) and the weak property (ω_1) of M_C .