Unbounded Products of Operators and Connections to Dirac-Type Operators

TL;DR

This paper investigates properties of unbounded operator products in Hilbert spaces, establishing new bounds and relations among their adjoints, with implications for Dirac operators and operator self-adjointness.

Contribution

It introduces new inclusion bounds and relations for products of unbounded operators, enhancing understanding of their adjoints and self-adjointness conditions.

Findings

Derived new inclusion bounds for operator product closures.

Established relations among operator adjoints for unbounded operators.

Clarified conditions for self-adjointness and normality of operator products.

Abstract

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 $\bar{A} \bar{B}$ and $\bar{AB}$ 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.