# On stability of closedness and self-adjointness for 2 × 2 operator matrices

Research paper by **A. A. Shkalikov, C. Trunk**

Indexed on: **19 Jan '17**Published on: **01 Nov '16**Published in: **Mathematical Notes**

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#### Abstract

Consider an operator which is defined in Banach or Hilbert space X = X
1 × X
2 by the matrix
\(L = \left( {\begin{array}{*{20}{c}}A&B \\ C&D \end{array}} \right)\)
, where the linear operators A: X
1 → X
1, B: X
2 → X
1, C: X
1 → X
2, and D: X
2 → 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 L. For the operator L acting in a Hilbert space, analogs of Rellich–Kato theorems on the stability of self-adjointness are obtained. Consider an operator which is defined in Banach or Hilbert space X = X
1 × X
2 by the matrix
\(L = \left( {\begin{array}{*{20}{c}}A&B \\ C&D \end{array}} \right)\)
, where the linear operators A: X
1 → X
1, B: X
2 → X
1, C: X
1 → X
2, and D: X
2 → 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 L. For the operator L acting in a Hilbert space, analogs of Rellich–Kato theorems on the stability of self-adjointness are obtained.XX1X2
\(L = \left( {\begin{array}{*{20}{c}}A&B \\ C&D \end{array}} \right)\)
\(L = \left( {\begin{array}{*{20}{c}}A&B \\ C&D \end{array}} \right)\)X1X1X2X1X1X2X2X2CBADLL