Some Selfadjoint 2 * 2 Operator Matrices Associated with Closed Operators

We study a 2 *2 operator matrix associated with a closed densely defined operator. Among others, the selfadjointness of a closed symmetric operator and the strong commutativity of two (unbounded) self-adjoint operators are characterized in terms of the related operator matrices. We propose a definition of strong commutativity for closed symmetric operators.