On commutators in algebras of unbounded operators
For a dense linear manifold D in a separable Hilbert space H, the set \(L^+(D)=\{A:AD\subset D\) and A*D\(\subset D\}\) forms a topological *- algebra of (unbounded) operators with the uniform topology given by the seminorms \(A\to \| A\|_ M=\sup_{x,y\in M}| <x,Ay>| (M\subset D\) t-bounded) [\textit{G. Lassner}, Rep. Math. Phys. 3, 279-293 (1972; Zbl 0252.46087)]. When D has the form \(D=D(T)=\cap_{n\geq 0}D(T^ n)\), where \(T=T^*\geq I\) is a selfadjoint diagonal operator, the author gives several criteria which imply diagonal, quasi-diagonal and finite dimensional operators in \(L^+(D)\) are commutators, and shows that the commutators are dense in \(L^+(D)\).
topological *-algebra of (unbounded) operators
scientific article; zbMATH DE number 4071734
scientific article