# 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 $0$.

**DOI: **http://dx.doi.org/10.7900/jot.2021dec12.2367

**Keywords: **Fuglede property, integral operator, Dixmier trace

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