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arXiv · 2605.19239

Weyl's laws and Connes' Trace Theorem for operator-valued pseudo-differential operators

Edward McDonald·Xiao Xiong·Xinyu Zhang
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Abstract

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.

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Edward McDonald, Xiao Xiong, Xinyu Zhang. 2026-05-19. Weyl's laws and Connes' Trace Theorem for operator-valued pseudo-differential operators. https://arxiv.org/abs/2605.19239

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