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The multiplicative anomaly for determinants of elliptic operators. (English) Zbl 0851.58048
The author, among other things, derives two new expressions for \[ \log \text{det } AB - \log \text{det } A - \log \text{det }B\tag \(*\) \] for suitable operators \(A\) and \(B\). When \(A\) and \(B\) commute, a formula for \((*)\) was computed by M. Wodzicki [Lect. Notes Math. 1289, 320-399 (1987; Zbl 0649.58033)].
In this paper one can find some interesting variations and generalizations of \((*)\) in the spirit of Analysis and Geometry.

MSC:
58J52 Determinants and determinant bundles, analytic torsion
58J40 Pseudodifferential and Fourier integral operators on manifolds
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[1] D. Burghelea, L. Friedlander, and T. Kappeler, On the determinant of elliptic differential and finite difference operators in vector bundles over \(S^1\) , Comm. Math. Phys. 138 (1991), no. 1, 1-18. · Zbl 0734.58043 · doi:10.1007/BF02099666
[2] T. Branson and B. Ørsted, Conformal geometry and global invariants , Differential Geom. Appl. 1 (1991), no. 3, 279-308. · Zbl 0785.53025 · doi:10.1016/0926-2245(91)90004-S
[3] T. Branson and B. Ørsted, Explicit functional determinants in four dimensions , Proc. Amer. Math. Soc. 113 (1991), no. 3, 669-682. JSTOR: · Zbl 0762.47019 · doi:10.2307/2048601 · links.jstor.org
[4] T. Branson, S. Y. A. Chang, and P. Yang, Estimates and extremals for zeta function determinants on four-manifolds , Comm. Math. Phys. 149 (1992), no. 2, 241-262. · Zbl 0761.58053 · doi:10.1007/BF02097624
[5] S. Y. A. Chang and P. Yang, Extremal metrices of zeta function determinants on \(4\)-manifolds , to appear in Annals of Math. JSTOR: · Zbl 0842.58011 · doi:10.2307/2118613 · links.jstor.org
[6] L. Friedlander, Determinants of Elliptic Operators , Thesis, Massachusetts Institute of Technology, 1989.
[7] S. Fulling and G. Kennedy, The resolvent parametrix of the general elliptic linear differential operator: a closed form for the intrinsic symbol , Trans. Amer. Math. Soc. 310 (1988), no. 2, 583-617. · Zbl 0711.35155 · doi:10.2307/2000982
[8] V. Guillemin, A new proof of Weyl’s formula on the asymptotic distribution of eigenvalues , Adv. in Math. 55 (1985), no. 2, 131-160. · Zbl 0559.58025 · doi:10.1016/0001-8708(85)90018-0
[9] L. Hörmander, The Analysis of Linear Partial Differential Operators III , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 274, Springer-Verlag, Berlin, 1985. · Zbl 0601.35001
[10] M. Kontseivick and S. Vishik, Determinants of elliptic pseudodifferential operators , preprint, 1994.
[11] I. Gohberg and M. Kreĭ n, Introduction to the theory of linear nonselfadjoint operators , Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. · Zbl 0181.13504
[12] K. Okikiolu, The Campbell-Hausdorff theorem for elliptic operators and a related trace formula , Duke Math. J. 79 (1995), no. 3, 687-722. · Zbl 0854.35137 · doi:10.1215/S0012-7094-95-07918-6
[13] A. Polyakov, Quantum geometry of bosonic strings , Phys. Lett. B 103 (1981), no. 3, 207-210. · doi:10.1016/0370-2693(81)90743-7
[14] A. Polyakov, Quantum geometry of Fermionic strings , Phys. Lett. B 103 (1981), no. 3, 211-213. · doi:10.1016/0370-2693(81)90744-9
[15] D. Ray and I. Singer, \(R\)-torsion and the Laplacian on Riemannian manifolds , Advances in Math. 7 (1971), 145-210. · Zbl 0239.58014 · doi:10.1016/0001-8708(71)90045-4
[16] V. Romanov and A. Schwartz, Anomalies and elliptic operators , Theor. and Math. Phys. 416 (1979), 190-204 (Russian).
[17] B. Osgood, R. Phillips, and P. Sarnak, Extremals of determinants of Laplacians , J. Funct. Anal. 80 (1988), no. 1, 148-211. · Zbl 0653.53022 · doi:10.1016/0022-1236(88)90070-5
[18] R. Seeley, Complex powers of an elliptic operator , Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), Proc. Sympos. Pure Math., vol. 10, Amer. Math. Soc., Providence, R.I., 1967, pp. 288-307. · Zbl 0159.15504
[19] R. Strichartz, A functional calculus for elliptic pseudodifferential operators , Amer. J. Math. 94 (1972), 711-722. JSTOR: · Zbl 0246.35082 · doi:10.2307/2373753 · links.jstor.org
[20] F. Treves, Introduction to pseudodifferential and Fourier integral operators. Vol. 1 , Plenum Press, New York, 1980. · Zbl 0453.47027
[21] M. Wodzicki, Non-commutative residue, Chapter I. Fundamentals , \(K\)-Theory, Arithmetic, and Geometry (Moscow, 1984-1986), Lecture Notes in Math, vol. 1289, Springer-Verlag, Berlin, 1987, pp. 320-399. · Zbl 0649.58033 · doi:10.1007/BFb0078372
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