Criteria of mutual adjointness of proper extensions of linear relations

In the paper the role of an initial object is played by a couple $(L,L_0)$ of closed linear relations in a Hilbert space $H$, such that $L_0 subset L$. Each closed linear relation $L_1(M_1)$ such that $L_0 subset L_1 subset L$ (respectively $L^{ast} subset M_1 subset L_0^{ast} $) is said to be a proper extension of $ L_0(L^{ast})$. In the terms of abstract boundary operators i.e. bounded linear operator $U(V)$ acting from $L(M)$ to $G$ ($G$ is an auxiliary Hilbert space) such that the null space of $U(V)$ contains $L_0(L^{ast})$, criteria of mutual adjointness for mentioned above relations $L_1$ and $M_1$ are established.