. The closed range and Fredholm properties of the upper-triangular operator matrix M = ( A C 0 B ) ∈ B ( H 1 ⊕ H 2 ) are studied, where H 1 and H 2 are Hilbert spaces. It is shown that the range R ( M ) of M is closed if and only if the following statements hold: where P G denotes the orthogonal projection onto G along G ⊥ . Moreover, the analogues for the Fredholmness of M are further presented.
