Weyl's laws and Connes' Trace Theorem for operator-valued pseudo-differential operators
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arXiv
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| Format: | Preprint |
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2026
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| author | McDonald, Edward Xiong, Xiao Zhang, Xinyu |
| author_facet | McDonald, Edward Xiong, Xiao Zhang, Xinyu |
| contents | We investigate the spectral asymptotic behavior of operator-valued classical pseudo-differential operators ($Ψ$DOs) for negative order with symbols taking values in a semifinite von Neumann algebran $\mathcal{M}$ equipped with a normal semifinite faithful trace. Within the framework of Connes' noncommutative geometry, we extend Connes' trace theorem to this operator-valued (type II) setting. Our main results are as follows: (i) a symbolic characterization of complex powers for operator-valued elliptic $Ψ$DOs, extending Seeley's classical construction; (ii) a trace formula for localized Riemann $ζ$-functions that links the spectral residues of operator-valued elliptic operators to their principal symbols, thereby providing an operator-valued extension of the Connes--Wodzicki residue; (iii) Weyl's law for right-compactly supported operator-valued classical $Ψ$DOs of arbitrary negative order, which yields a direct spectral proof of the noncommutative integral that bypasses the use of Dixmier traces; (iv) Weyl's law for operator-valued commutators of certain Fourier multipliers with multiplication operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_19239 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Weyl's laws and Connes' Trace Theorem for operator-valued pseudo-differential operators McDonald, Edward Xiong, Xiao Zhang, Xinyu Operator Algebras 46L51, 58B34, 47G30, 35P20 We investigate the spectral asymptotic behavior of operator-valued classical pseudo-differential operators ($Ψ$DOs) for negative order with symbols taking values in a semifinite von Neumann algebran $\mathcal{M}$ equipped with a normal semifinite faithful trace. Within the framework of Connes' noncommutative geometry, we extend Connes' trace theorem to this operator-valued (type II) setting. Our main results are as follows: (i) a symbolic characterization of complex powers for operator-valued elliptic $Ψ$DOs, extending Seeley's classical construction; (ii) a trace formula for localized Riemann $ζ$-functions that links the spectral residues of operator-valued elliptic operators to their principal symbols, thereby providing an operator-valued extension of the Connes--Wodzicki residue; (iii) Weyl's law for right-compactly supported operator-valued classical $Ψ$DOs of arbitrary negative order, which yields a direct spectral proof of the noncommutative integral that bypasses the use of Dixmier traces; (iv) Weyl's law for operator-valued commutators of certain Fourier multipliers with multiplication operators. |
| title | Weyl's laws and Connes' Trace Theorem for operator-valued pseudo-differential operators |
| topic | Operator Algebras 46L51, 58B34, 47G30, 35P20 |
| url | https://arxiv.org/abs/2605.19239 |