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.
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Translated from Neliniini Kolyvannya, Vol. 26, No. 1, pp. 42–54, January–March, 2023.
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Zhuravlev, V.P., Fomin, M.P. Generalized Inversion of the Operator Matrices (Generalization of the Frobenius Theorem). J Math Sci 278, 974–987 (2024). https://doi.org/10.1007/s10958-024-06975-8
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DOI: https://doi.org/10.1007/s10958-024-06975-8