Generalized Inversion of the Operator Matrices (Generalization of the Frobenius Theorem)

We propose methods for the construction of the operator matrices generalized inverse to (2 × 2) operator matrices in Banach spaces. We obtain the condition of solvability and a formula for representation of at least one solution of the operator equation in terms of a (2 × 2) operator matrix. It is shown that the obtained formula for the generalized inverse operator matrix is related to the well-known Frobenius formula for the construction of the inverse matrix for a nondegenerate block matrix.