TY - JOUR

T1 - Noncommutative Residues, Equivariant Traces, and Trace Expansions for an Operator Algebra on Rn

AU - Savin, Anton

AU - Schrohe, Elmar

PY - 2024/4/24

Y1 - 2024/4/24

N2 - We consider an algebra \(\mathscr A\) of Fourier integral operators on \(\mathbb R^n\). It consists of all operators \(D: \mathscr S(\mathbb R^n)\to \mathscr S(\mathbb R^n)\) on the Schwartz space \(\mathscr S(\mathbb R^n)\) that can be written as finite sums \(\) D= \sum R_gT_w A, \(\) with Shubin type pseudodifferential operators \(A\), Heisenberg-Weyl operators \(T_w\), \(w\in \mathbb C^n\), and lifts \(R_g\), \(g\in \mathrm U(n)\), of unitary matrices \(g\) on \(\mathbb C^n\) to operators \(R_g\) in the complex metaplectic group. For \(D \in \mathscr A\) and a suitable auxiliary Shubin pseudodifferential operator \(H\) we establish expansions for \(\mathop{\mathrm {Tr}}(D(H-\lambda)^{-K})\) as \(|\lambda| \to \infty\) in a sector of \(\mathbb C\) for sufficiently large \(K\) and of \(\mathop{\mathrm {Tr}}(De^{-tH})\) as \(t\to 0^+\). We also obtain the singularity structure of the meromorphic extension of \(z\mapsto \mathop{\mathrm{Tr}}(DH^{-z})\) to \(\mathbb C\). Moreover, we find a noncommutative residue as a suitable coefficient in these expansions and construct from it a family of localized equivariant traces on the algebra.

AB - We consider an algebra \(\mathscr A\) of Fourier integral operators on \(\mathbb R^n\). It consists of all operators \(D: \mathscr S(\mathbb R^n)\to \mathscr S(\mathbb R^n)\) on the Schwartz space \(\mathscr S(\mathbb R^n)\) that can be written as finite sums \(\) D= \sum R_gT_w A, \(\) with Shubin type pseudodifferential operators \(A\), Heisenberg-Weyl operators \(T_w\), \(w\in \mathbb C^n\), and lifts \(R_g\), \(g\in \mathrm U(n)\), of unitary matrices \(g\) on \(\mathbb C^n\) to operators \(R_g\) in the complex metaplectic group. For \(D \in \mathscr A\) and a suitable auxiliary Shubin pseudodifferential operator \(H\) we establish expansions for \(\mathop{\mathrm {Tr}}(D(H-\lambda)^{-K})\) as \(|\lambda| \to \infty\) in a sector of \(\mathbb C\) for sufficiently large \(K\) and of \(\mathop{\mathrm {Tr}}(De^{-tH})\) as \(t\to 0^+\). We also obtain the singularity structure of the meromorphic extension of \(z\mapsto \mathop{\mathrm{Tr}}(DH^{-z})\) to \(\mathbb C\). Moreover, we find a noncommutative residue as a suitable coefficient in these expansions and construct from it a family of localized equivariant traces on the algebra.

KW - math.OA

KW - math.FA

KW - 58J40, 58J42

U2 - 10.48550/arXiv.2303.14171

DO - 10.48550/arXiv.2303.14171

M3 - Article

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

ER -