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

TL;DR

This paper extends classical spectral asymptotics and Connes' trace theorem to operator-valued pseudo-differential operators within noncommutative geometry, providing new trace formulas and Weyl's laws.

Contribution

It introduces a symbolic characterization of complex powers, trace formulas for spectral residues, and Weyl's laws for operator-valued pseudo-differential operators, advancing noncommutative spectral analysis.

Findings

Extended Connes' trace theorem to operator-valued setting

Derived Weyl's law for operator-valued pseudo-differential operators

Established trace formulas linking spectral residues to principal symbols

Abstract

We investigate the spectral asymptotic behavior of operator-valued classical pseudo-differential operators ($\Psi$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 $\Psi$DOs, extending Seeley's classical construction; (ii) a trace formula for localized Riemann $\zeta$-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 $\Psi$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.