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

Abstract

Consider an operator which is defined in Banach or Hilbert space X = X₁ × X₂ by the matrix \(L = \left( {\begin{array}{*{20}{c}}A&B \\ C&D \end{array}} \right)\), where the linear operators A: X₁ → X₁, B: X₂ → X₁, C: X₁ → X₂, and D: X₂ → X₂ 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.