Some results concerning the linear relations which are adjoint to each other

In order to study the stability of self-adjointness of a block linear relation matrix with unbounded entries acting in Hilbert spaces, we start by discussing the question whether a linear relation is identical with the adjoint of another linear relation. After that, based on the assumptions obtained, we provide other necessary and sufficient conditions about the entries for a linear relation matrix to be self-adjoint. Our results generalize one of the most important known results in the literature that works on the self-adjointness for linear relations (see Shi et al. in Linear Algebra Appl 438(1):191-218, 2013, Linear Multilinear Algebra 66(2):309-333, 2018; Shi and Xu in Linear Algebra Appl 531:547-574, 2017).