The semi-Fredholmness of 2×2 operator matrices

Let H1 and H2 be separable infinite-dimensional Hilbert spaces, and let A∈B(H1), B∈B(H2) and C∈B(H2,H1) be given operators. A necessary and sufficient condition is obtained for to be a right (left) Fredholm operator for some X∈B(H1,H2).