Some Fredholm left invertible completion problems for operator matrices

Abstract

The structure of the weak approximate spectrum of two-by-two upper triangular operator matrix MC  acting on a Hilbert space H ⊕ K is studied. First, we characterize the relationship between the specturm σ(MC ) and the weak approximate spectrum σF σa (MC ) and give the equivalent conditions that makeσF σa (MC ) = σ(MC ) and σF a (MC ) = σF a (A) ∪ σF a (B)according to the properties that the operators A and B satisfy. Then,we study the weak property (ω1) of MC and explore the relationshipbetween σ(MC ) = σF a (MC ) and the weak property (ω1) of MC .