We first show that a derivation studied recently by E. Christensen has a set of analytic elements which is strong operator topology-dense in the algebra of bounded operators on a Hilbert space, which strengthens a result of Christensen. Our second main result shows that this derivation has kernel stabilization, that is, no elements have derivative eventually equal to 0 unless their first derivative is 0. As applications, we (1) show that a family of derivations on C*-algebras studied by Bratteli and Robinson has kernel stabilization, and (2) we provide sufficient conditions for when two operators which satisfy the Heisenberg Commutation Relation must both be unbounded.

The results in this paper appear in the author's doctoral dissertation, written under the supervision of A. Donsig and D. Pitts at the University of Nebraska-Lincoln. We would like to thank Donsig and Pitts for countless research conversations and their invaluable guidance in the crafting of this paper. We are grateful for the referee's suggested revisions and insightful comments. In addition, we would like to thank Magnus Goffeng for his expertise in unbounded operator theory, Nik Weaver for his thoughtful comments in the late versions of the paper, and Ruy Exel for suggesting the Riesz map as a remedy for the analytic element density argument. Last, the author would like to dedicate this paper to Professor Anthony F. Starace (7/24/1945 - 9/5/2019) in memory of his eloquent instruction of quantum mechanics and with gratitude for his connecting the author to related work within the physics community.