The Fuglede commutativity theorem modulo operator ideals

Abstract:

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