Fuglede theorem for symmetric spaces of $\tau$-measurable operators

TL;DR

This paper extends the Fuglede theorem to symmetric operator ideals, characterizing when such spaces satisfy the theorem based on their Boyd indices and interpolation properties, unifying previous cases like Lorentz and Schatten classes.

Contribution

It provides a complete characterization of symmetric ideals satisfying the Fuglede theorem, linking it to Boyd indices and interpolation space properties.

Findings

Characterization of symmetric ideals satisfying Fuglede theorem

Connection between Boyd indices and Fuglede theorem validity

Unification of known cases like Lorentz and Schatten classes

Abstract

We extend the classical Fuglede commutativity theorem to the full scale of symmetrically normed operator ideals. Our main result provides a complete characterization: a symmetric ideal or symmetric operator space of $\tau$-measurable operators satisfies the Fuglede theorem if and only if its commutative core has non-trivial Boyd indices, or equivalently, if it is an interpolation space in the scale of $L_p$-spaces for $1<p<\infty$. This criterion subsumes all previously known cases, including Lorentz and Schatten classes.