Traces and Residues of Pseudo-Differential Operators on the Torus

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• 作者：Albrecht Pietsch
• 关键词：Primary 47B10 ; 35S05 ; Secondary 46B45 ; Trace ; Operator ideal ; Dyadic representation ; Shift ; invariant linear form ; Pseudo ; differential operator ; Symbol ; Residue ; Connes-trace theorem
• 刊名：Integral Equations and Operator Theory
• 出版年：2015
• 出版时间：September 2015
• 年：2015
• 卷：83
• 期：1
• 页码：1-23
• 全文大小：619 KB
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  13.Pietsch A.: Traces on operator ideals and related linear forms on sequence ideals (part III). J. Math. Anal. Appl. 421, 971-81 (2015)MATH MathSciNet CrossRef
  14.Pietsch, A.: Traces on Operator Ideals and Related Linear Forms on Sequence Ideals (part IV) (in preparation)
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  16.Saranen J., Vainikko G.: Periodic Integral and Pseudodifferential Equations with Numerical Application. Springer, Berlin (2002)CrossRef
  17.Semenov, E., Sukochev, F., Usachev, A., Zanin, D.: Banach limits and traces on \({\mathscr{L}_{1, \infty}}\) . Adv. Math. (submitted)
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  19.Wodzicki M.: Noncommutative residue. Lect. Notes Math. 1289, 320-99 (1987)MathSciNet CrossRef
  
• 作者单位：Albrecht Pietsch (1)
  
  1. Biberweg 7, 07749, Jena, Germany
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  
• 出版者：Birkh盲user Basel
• ISSN：1420-8989

文摘

This paper is the outcome of an attempt to understand the connection between singular traces and the Wodzicki residues of pseudo-differential operators on closed Riemannian manifolds as presented in the recent monograph of Lord, Sukochev, and Zanin. Employing my technique of dyadic representations of operators, I am able to replace the mountain tour performed by Sukochev and his coauthors through a walk in a park. The crucial point is that considerations about eigenvalues are no longer involved. To simplify understanding, the new approach is demonstrated by the example of pseudo-differential operators on the d-dimensional flat torus \({\mathbb{T}^d}\). In this special case it is possible to work with global symbols (which need not be smooth). Keywords Trace Operator ideal Dyadic representation Shift-invariant linear form Pseudo-differential operator Symbol Residue Connes-trace theorem