On Stability of Closedness and Self-Adjointness for $2\times 2$ Operator Matrices

Consider an operator which is defined in Banach or Hilbert space $X=X_1\times X_2$ by the matrix
\begin{equation*} \mathbf L = \begin{pmatrix} A & B \\ C & D \end{pmatrix}, \end{equation*}
where the linear operators $A\colon X_1 \to X_1$, $B\colon X_2 \to X_1$, $C\colon X_1\to X_2$, and $D\colon X_2\to X_2$ are assumed to be unbounded. In the case when the operators $C$ and $B$ are relatively bounded with respect to the operators $A$ and $D$, respectively, new conditions of closedness or closability are obtained for the operator $\mathbf L$. For the operator $\mathbf L$ acting in a Hilbert space, analogs of Rellich–Kato theorems on the stability of self-adjointness are obtained.