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<p>The famous Helton-Howe trace formula was originally established for antisymmetric sums of Toeplitz operators on the Bergman space of the unit ball. This formula has been extended to weighted Bergman spaces and to the Hardy space in [<CitationRef CitationID="CR20">20</CitationRef>], and more recently it even has been extended to the Drury-Arveson space [<CitationRef CitationID="CR24">24</CitationRef>]. In this paper we show that an analogue of the Helton-Howe trace formula also holds in an unlikely place, the Fock space! We further show that our techniques not only work for the ordinary trace <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\text {tr}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mtext>tr</mtext> </math></EquationSource> </InlineEquation>, as in the case of the Helton-Howe trace formula, but also for the Dixmier trace.</p>

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The Helton-Howe Trace Formula for the Fock Space

  • 期刊论文

  • 发表日期：

    2026年7月3日

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  • Qinghua Hu,
  • Jingbo Xia

摘要

The famous Helton-Howe trace formula was originally established for antisymmetric sums of Toeplitz operators on the Bergman space of the unit ball. This formula has been extended to weighted Bergman spaces and to the Hardy space in [20], and more recently it even has been extended to the Drury-Arveson space [24]. In this paper we show that an analogue of the Helton-Howe trace formula also holds in an unlikely place, the Fock space! We further show that our techniques not only work for the ordinary trace \({\text {tr}}\) tr , as in the case of the Helton-Howe trace formula, but also for the Dixmier trace.