The Fuglede commutativity theorem modulo operator ideals

Abstract

Let H H denote a separable, infinite-dimensional complex Hilbert space. A two-sided ideal I I of operators on H H possesses the generalized Fuglede property (GFP) if, for every normal operator N N and every X ∈ L ( H ) X \in L(H) , N X − X N ∈ I NX - XN \in I implies N ∗ X − X N ∗ ∈ I {N^ * }X - X{N^ * } \in I . Fuglede’s Theorem says that I = { 0 } I = \left \{ 0 \right \} has the GFP. It is known that the class of compact operators and the class of Hilbert-Schmidt operators have the GFP. We prove that the class of finite rank operators and the Schatten p p -classes for 0 > p > 1 0 > p > 1 fail to have the GFP. The operator we use as an example in the case of the Schatten p p -classes is multiplication by z + w z + w on L 2 {L^2} of the torus.