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

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

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

#### 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