Unbounded derivations on B(H)

A derivation on a C*-algebra is inner if it is defined as the commutator with a fixed (bounded) element of that C*-algebra. In physical systems, however, the element which implements a derivation via a commutator is unbounded, which presents challenges in studying the derivation’s properties and its higher powers. In 2017, E. Christensen introduced and studied a weakly-defined derivation on B(H) which is defined as a commutator with a fixed unbounded self-adjoint operator. E. Christensen’s definition is very nice in the sense that we need only study differentiability of a complex-valued function rather than definedness of a commutator involving an unbounded operator. 

In this talk, we review the relationship between a derivation and the one-parameter automorphism group for which it is the infinitesimal generator. We then consider the density of the domains of higher powers of the weakly-defined derivation on B(H) introduced by E. Christensen, and we prove that the analytic vectors for this derivation is SOT-dense. We also study and prove that derivations of this form have a surprising property called kernel stabilization. As a consequence of this property, we provide new sufficient conditions for when two operators which satisfy the Heisenberg Commutation Relation must both be unbounded.