The semi-Fredholmness of operator matrices

Let H 1 and H 2 be separable infinite-dimensional Hilbert spaces, and let A ∈ B ( H 1 ) , B ∈ B ( H 2 ) and C ∈ B ( H 2 , H 1 ) be given operators. A necessary and sufficient condition is obtained for ( A C X B ) to be a right (left) Fredholm operator for some X ∈ B ( H 1 , H 2 ) .