Finite operators

Abstract

A bounded linear operator A A on a Hilbert space H H is called finite if | | A X − X A − 1 | | ≧ 1 ||AX - XA - 1|| \geqq 1 for each X ∈ B ( H ) X \in B(H) . The class of finite operators is uniformly closed, contains every normal operator, every operator with a compact direct summand, and the entire C ∗ {C^ \ast } -algebra generated by each of its members. These results imply that the set of operators with a finite dimensional reducing subspace is not uniformly dense. It is also shown that the set of self-commutators is uniformly closed.