Unbounded products of operators and connections to Dirac-type operators

Let A and B be two densely defined unbounded closeable operators in a Hilbert space such that their unbounded operator products AB and BA   are also densely defined. Then all four operators possess adjoints and we obtain new inclusion bounds for the operator product closures A¯B¯ and AB¯ in terms of new relations among the operator adjoints. These in turn lead to sharpened understandings for when products of unbounded self-adjoint and unbounded normal operators are self-adjoint and normal. They also clarify certain operator-product issues for Dirac operators.