Hermitian treatment of Dyson boson theory

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Abstract
A great merit of the Dyson boson mapping theory is in its finiteness of boson expansion, but the demerit is in the non-Hermiticity of the Hamiltonian in the boson space. This demerit can be overcome by using the Hermitian treatment proposed in a previous paper. Given here is a precise proof of the method, by which the insufficient explanation in the previous paper is supplemented and the limit of applicability of the treatment is clarified.