Osaka Journal of Mathematics

Zeta determinant and operator determinants

Mauro Spreafico

Full-text: Open access

Abstract

We apply techniques of zeta functions and regularized products theory to study the zeta determinant of a class of abstract operators with compact resolvent, and in particular the relation with other spectral functions.

Article information

Source
Osaka J. Math., Volume 48, Number 1 (2011), 41-50.

Dates
First available in Project Euclid: 22 March 2011

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1300802703

Mathematical Reviews number (MathSciNet)
MR2802591

Zentralblatt MATH identifier
1222.58028

Subjects
Primary: 58J52: Determinants and determinant bundles, analytic torsion
Secondary: 58J40: Pseudodifferential and Fourier integral operators on manifolds [See also 35Sxx] 11M36: Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas

Citation

Spreafico, Mauro. Zeta determinant and operator determinants. Osaka J. Math. 48 (2011), no. 1, 41--50. https://projecteuclid.org/euclid.ojm/1300802703


Export citation

References

  • J.R. Quine and J. Choi: Zeta regularized products and functional determinants on spheres, Rocky Mountain J. Math. 26 (1996), 719–729.
  • L. Friedlander: The asymptotics of the determinant function for a class of operators, Proc. Amer. Math. Soc. 107 (1989), 169–178.
  • P.B. Gilkey: Invariance Theory, the Heat Equation, and the Atiyah–Singer Index Theorem, second edition, Studies in Advanced Mathematics, CRC, Boca Raton, FL, 1995.
  • I.C. Gohberg and M.G. Kreĭ n: Introduction to the Theory of Linear Nonselfadjoint Operators, Translations of Mathematical Monographs 18, Amer. Math. Soc., Providence, RI, 1969.
  • J. Jorgenson and S. Lang: Basic Analysis of Regularized Series and Products, Lecture Notes in Mathematics 1564, Springer, Berlin, 1993.
  • D.B. Ray and I.M. Singer: $R$-torsion and the Laplacian on Riemannian manifolds, Advances in Math. 7 (1971), 145–210.
  • P. Sarnak: Determinants of Laplacians, Comm. Math. Phys. 110 (1987), 113–120.
  • R.T. Seeley: Complex powers of an elliptic operator; in Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), Amer. Math. Soc., Providence, RI, 1967, 288–307.
  • M. Spreafico: Zeta functions and regularized determinants on projective spaces, Rocky Mountain J. Math. 33 (2003), 1499–1512.
  • M. Sprefiko: A generalization of the Euler gamma function, Funktsional. Anal. i Prilozhen. 39 (2005) 87–91.
  • M. Spreafico: Zeta invariants for sequences of spectral type, special functions and the Lerch formula, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006), 863–887.
  • A. Voros: Spectral functions, special functions and the Selberg zeta function, Comm. Math. Phys. 110 (1987), 439–465.