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Block basis property of a class of 2 × 2 operator matrices and its application to elasticity

A necessary and sufficient condition is obtained for the generalized eigenfunction systems of 2 × 2 operator matrices to be a block Schauder basis of some Hilbert space, which offers a mathematical foundation of solving symplectic elasticity problems by using the method of separation of variables. Moreover, the theoretical result is applied to two plane elasticity problems via the separable Hamiltonian systems.