This paper concerns the discretizations in pair of adjoint operators between Hilbert spaces so that the adjoint properties can be preserved. Due to the finite-dimensional essence of discretized operators, a new framework, theory of partially adjoint operators, is motivated and presented in this paper, so that adjoint properties can be figured out for finite-dimensional operators which can not be non-trivially densely defined in other background spaces. A formal methodology is presented under the framework to construct partially adjoint discretizations by a conforming discretization (CD) and an accompanied-by-conforming discretization (ABCD) for each of the operators. Moreover, the methodology leads to an asymptotic uniformity of an infinite family of finite-dimensional operators. The validities of the theoretical framework and the formal construction of discretizations are illustrated by a systematic family of in-pair discretizations of the adjoint exterior differential operators. The adjoint properties concerned in the paper are the closed range theorem and the strong dualities, whose preservations have not been well studied yet. Quantified versions of the closed range theorem are established for both adjoint operators and partially adjoint discretizations. The notion Poincare-Alexander-Lefschetz (P-A-L for short) type duality is borrowed for operator theory, and horizontal and vertical P-A-L dualities are figured out for adjoint operators and their analogues are established for partially adjoint discretizations. Particularly by partially adjoint discretizations of exterior differential operators, the Poincare-Lefschetz duality is preserved as an identity, which was not yet obtained before.

翻译：本文涉及Hilbert空间间对口操作员的离心操作员的离心操作员,这样可以保存离心操作员的离心操作员的离心操作员。由于离心操作员的有限层面精髓,本文件提出并介绍了一个新的框架,即部分离心操作员的理论理论,这样就可以为其他背景空间中无法非分心定义的离心操作员找到离心操作员的离心特性。在框架之下提出了一个正式的方法,通过符合离心操作员的离心操作员(CD)和对每个操作员的离心离心操作员的离心操作员的离心操作员(ABCD)来建立部分离心操作员的离心操作。 此外,对离心操作员的离心操作员的离心操作员的离心操作员和双尾离心操作员的离心操作员的离心操作员的离心操作员是内部离心操作员。