×

zbMATH — the first resource for mathematics

Spectral asymmetry, zeta functions, and the noncommutative residue. (English) Zbl 1127.58030
Given a sector \(\Lambda\) of the complex plane and a pseudodifferential operator \(P\) whose principal symbol does not have eigenvalues in \(\Lambda\), one can construct complex powers of \(P\) using well-chosen cuts in the complex plane along rays in \(\Lambda\). These form analytic families of pseudodifferential operators, and their traces are meromorphic functions with at most simple poles at certain reals. The present paper deals with regularity issues of such zeta functions and of their differences when considering various cuts. This has been initially studied by M. Wodzicki in [Invent. Math. 66, 115–135 (1982; Zbl 0489.58030)]. Here Ponge treats the more refined issue of odd operators (which include differential operators and their parametrices). He proves, among other things, that on an odd-dimensional manifold, the zeta function of an even-order differential operator (with respect to a cut as above) is regular at all integers, and its values there are independent of the cut. He also recovers a result of G. Grubb, stating that the eta function of a self-adjoint first-order differential operator on an even-dimensional manifold is entire. This had been proved by Branson and Gilkey for compatible Dirac operators.

MSC:
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58J42 Noncommutative global analysis, noncommutative residues
58J40 Pseudodifferential and Fourier integral operators on manifolds
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.1002/cpa.3160150203 · Zbl 0109.32701 · doi:10.1002/cpa.3160150203
[2] Agranovich M. S., Encyclopaedia of Mathematical Sciences 63, in: Partial Differential Equations. VI (1994) · doi:10.1007/978-3-662-09209-5_1
[3] Agranovich M. S., Z. Anal. Anwendungen 8 pp 237–
[4] DOI: 10.1017/S0305004100049410 · Zbl 0297.58008 · doi:10.1017/S0305004100049410
[5] DOI: 10.1017/S0305004100052105 · Zbl 0325.58015 · doi:10.1017/S0305004100052105
[6] DOI: 10.1007/978-3-642-58088-8 · doi:10.1007/978-3-642-58088-8
[7] J.M. Bismut, Surveys in Differential Geometry III (International Press, Boston, MA, 1998) pp. 1–76.
[8] Branson T., J. Funct. Anal. 108
[9] DOI: 10.1215/S0012-7094-99-09613-8 · Zbl 0956.58014 · doi:10.1215/S0012-7094-99-09613-8
[10] DOI: 10.2307/2374646 · Zbl 0664.58035 · doi:10.2307/2374646
[11] Burak T., Ann. Scuola Norm. Sup. Pisa (3) 22 pp 113–
[12] Burak T., Ann. Scuola Norm. Sup. Pisa (3) 24 pp 209–
[13] DOI: 10.1007/BF02506388 · Zbl 0881.58009 · doi:10.1007/BF02506388
[14] DOI: 10.1007/BF01895667 · Zbl 0960.46048 · doi:10.1007/BF01895667
[15] Dunford N., Linear Operators. Part II: Spectral Theory. Self Adjoint Operators in Hilbert Space (1963) · Zbl 0128.34803
[16] DOI: 10.1007/BF01364633 · Zbl 0405.58045 · doi:10.1007/BF01364633
[17] DOI: 10.1016/S0001-8708(81)80007-2 · Zbl 0469.58015 · doi:10.1016/S0001-8708(81)80007-2
[18] Gilkey P. B., Studies in Advanced Mathematics, in: Invariance Theory, the Heat Equation, and the Atiyah–Singer Index Theorem (1995)
[19] Grubb G., Contemporary Mathematics 366 pp 67– (2005) · doi:10.1090/conm/366/06725
[20] DOI: 10.1007/BF01884310 · Zbl 0851.58043 · doi:10.1007/BF01884310
[21] Gohberg I. C., Translations of Mathematical Monographs 18, in: Introduction to the Theory of Linear Nonselfadjoint Operators (1969)
[22] DOI: 10.1016/0001-8708(85)90018-0 · Zbl 0559.58025 · doi:10.1016/0001-8708(85)90018-0
[23] DOI: 10.1006/aima.1993.1064 · Zbl 0803.58052 · doi:10.1006/aima.1993.1064
[24] DOI: 10.1006/jfan.1993.1096 · Zbl 0791.35162 · doi:10.1006/jfan.1993.1096
[25] DOI: 10.1016/0393-0440(94)00032-Y · Zbl 0826.58008 · doi:10.1016/0393-0440(94)00032-Y
[26] DOI: 10.1007/BF02099890 · Zbl 0823.58046 · doi:10.1007/BF02099890
[27] DOI: 10.1007/978-3-642-66282-9 · doi:10.1007/978-3-642-66282-9
[28] DOI: 10.1007/978-1-4612-4262-8_6 · doi:10.1007/978-1-4612-4262-8_6
[29] Riesz F., Leçons d’Analyse Fonctionnelle (1952)
[30] DOI: 10.1081/PDE-200050102 · Zbl 1236.58037 · doi:10.1081/PDE-200050102
[31] R. T. Seeley, Singular Integrals, Proceedings of Symposia in Pure Mathematics 10 (American Mathematical Society, Providence, RI, Chicago, IL, 1976) pp. 288–307.
[32] DOI: 10.1080/03605308608820438 · Zbl 0598.35013 · doi:10.1080/03605308608820438
[33] DOI: 10.1007/978-3-642-96854-9 · Zbl 0616.47040 · doi:10.1007/978-3-642-96854-9
[34] DOI: 10.1007/BF01404760 · Zbl 0489.58030 · doi:10.1007/BF01404760
[35] DOI: 10.1007/BF01403095 · Zbl 0538.58038 · doi:10.1007/BF01403095
[36] Wodzicki M., Hermann Weyl’s Selected Papers (1985)
[37] DOI: 10.1007/BFb0078372 · doi:10.1007/BFb0078372
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.