Geodesic
On Stability of Closedness and Self-Adjointness for $2\times 2$ Operator Matrices
Matematičeskie zametki, Tome 100 (2016) no. 6, pp. 932-938

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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.
Keywords: operator matrices, perturbations of linear operators, closed operators, self-adjoint operators. 
@article{MZM_2016_100_6_a14,
     author = {A. A. Shkalikov and C. Trunk},
     title = {On {Stability} of {Closedness} and {Self-Adjointness} for $2\times 2$ {Operator} {Matrices}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {932--938},
     publisher = {mathdoc},
     volume = {100},
     number = {6},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2016_100_6_a14/}
}
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A. A. Shkalikov; C. Trunk. On Stability of Closedness and Self-Adjointness for $2\times 2$ Operator Matrices. Matematičeskie zametki, Tome 100 (2016) no. 6, pp. 932-938. http://geodesic.mathdoc.fr/item/MZM_2016_100_6_a14/