On the closedness and self-adjointness of 3 × 3 operator matrices and application to a non-relativistic three-channel potential scattering model

Abstract

In this paper, we are concerned with a $3\times 3$ block operator matrices acting in a Banach or Hilbert space ${\mathit{X}}_1\times {\mathit{X}}_2\times {\mathit{X}}_3$ given by $$\begin{array}{c}\left(\begin{array}{ccc}\hfill {\mathit{A}}_1\hfill & \hfill {\mathit{B}}_1\hfill & \hfill {\mathit{C}}_1\hfill \\ \hfill {\mathit{A}}_2\hfill & \hfill {\mathit{B}}_2\hfill & \hfill {\mathit{C}}_2\hfill \\ \hfill {\mathit{A}}_3\hfill & \hfill {\mathit{B}}_3\hfill & \hfill {\mathit{C}}_3\hfill \end{array}\right)\mathrm{,}\end{array}$$ where the linear entries are assumed to be unbounded. We study the closure as well as the self-adjointness in the case where the linear operators ${\mathit{A}}_2$ and ${\mathit{A}}_3$ are ${\mathit{A}}_1$ -bounded, ${\mathit{B}}_1$ and ${\mathit{B}}_3$ are ${\mathit{B}}_2$ -bounded and ${\mathit{C}}_1$ and ${\mathit{C}}_2$ are ${\mathit{C}}_3$ -bounded. These results are of importance to a non-relativistic three-channel potential scattering model.