The asymptotics of the determinant function for a class of operators

Friedlander, Leonid

doi:10.1090/S0002-9939-1989-0975642-0

The asymptotics of the determinant function for a class of operators

Author: Leonid Friedlander
Journal: Proc. Amer. Math. Soc. 107 (1989), 169-178
MSC: Primary 58G15; Secondary 47B25, 47B38, 47G05
DOI: https://doi.org/10.1090/S0002-9939-1989-0975642-0
MathSciNet review: 975642
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Abstract: Let be an elliptic pseudodifferential operator on a closed manifold and ${\text{ord}}A > \dim M$ . We derive the asymptotics of $\log \det (1 + \varepsilon {A^{ - 1}})$ when $\varepsilon \to \infty$ . The constant term of this asymptotics equals $- \log \det A$ .

[1] R. T. Seeley, Complex powers of an elliptic operator, Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966) Amer. Math. Soc., Providence, R.I., 1967, pp. 288–307. MR 0237943
[2] Michael E. Taylor, Pseudodifferential operators, Princeton Mathematical Series, vol. 34, Princeton University Press, Princeton, N.J., 1981. MR 618463
[3] M. A. Shubin, Pseudodifferential operators and spectral theory, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1987. Translated from the Russian by Stig I. Andersson. MR 883081
[4] N. N. Vakhania, Probability distributions on linear spaces, North-Holland Publishing Co., New York-Amsterdam, 1981. Translated from the Russian by I. I. Kotlarski; North-Holland Series in Probability and Applied Mathematics. MR 626346
[5] M. S. Agranovich, Some asymptotic formulas for elliptic pseudodifferential operators, Funktsional. Anal. i Prilozhen. 21 (1987), no. 1, 63–65 (Russian). MR 888015
[6] H. Bateman and A. Erdelyi, Higher transcendental functions, Vol. 1, McGraw-Hill Comp. Inc., New York-Toronto-London, 1953.