The Theory for J-Hermitian Subspaces in a Product Space

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This paper is concerned with the theory for J-Hermitian subspaces. The defect index of a J-Hermitian subspace is defined, and a formula for the defect index is established; the result that every J-Hermitian subspace has a J-self-adjoint subspace extension is obtained; all the J-self-adjoint subspace extensions of a J-Hermitian subspace are characterized. This theory will provide a fundamental basis for characterizations of J-self-adjoint extensions for linear nonsymmetric expressions on general time scales in terms of boundary conditions, including both differential and difference cases.