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

The Theory for J-Hermitian Subspaces in a Product Space