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We obtain asymptotic formulas for eigenvalues and eigenfunctions of theoperator generated by a system of ordinary differential equations with summablecoefficients and periodic or antiperiodic boundary conditions. Then using theseasymptotic formulas, we find necessary and sufficient conditions on thecoefficients for which the system of eigenfunctions and associated functions ofthe operator under consideration forms a Riesz basis.

On the Basis Property of the Root Functions of Differential Operators with Matrix Coefficients