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Journal of Operator Theory

Volume 90, Issue 2, Autumn 2023  pp. 401-423.

Integral operators associated wity the generalized Fuglede property

Authors:  Jingbo Xia
Author institution: Department of Mathematics, State University of New York at Buffalo, Buffalo, NY 14260, U.S.A.

Summary:  We consider a class of integral operators $T_{K,\mu }$ that arise from various problems studied in the past, one of which is the generalized Fuglede commutation property. We show that under a rather general growth condition on the measure $\mu $, the operator $T_{K,\mu }$ belongs to the Lorentz ideal $\mathcal{C}_1^+$. This naturally leads to the challenge of computing the Dixmier trace of $T_{K,\mu }$. In the case where $\mu $ is the restriction of the Lebesgue measure $m_n$ to a bounded Borel set in ${\mathbb R}^n$, we show that the Dixmier trace of $T_{K,\mu }$ is $0$.

Keywords: Fuglede property, integral operator, Dixmier trace

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