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Spectral Theory of Integral Operators

In: Basic Classes of Linear Operators

Author

Listed:
  • Israel Gohberg

    (Tel Aviv University, School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences)

  • Seymour Goldberg

    (University of Maryland, Department of Mathematics)

  • Marinus A. Kaashoek

    (Vrije Universiteit Amsterdam, Department of Mathematics and Computer Science)

Abstract

Using the theory developed in Chapter IV, we now present some fundamental theorems concerning the spectral theory of compact self adjoint integral operators. In general, the spectral series representations of these operators converge in the L2-norm which is not strong enough for many applications. Therefore we prove the Hilbert-Schmidt theorem and Mercer’s theorem since each of these theorems gives conditions for a uniform convergence of the spectral decomposition of the integral operators. As a corollary of Mercer’s theorem we obtain the trace formula for positive integral operators with continuous kernel function.

Suggested Citation

  • Israel Gohberg & Seymour Goldberg & Marinus A. Kaashoek, 2003. "Spectral Theory of Integral Operators," Springer Books, in: Basic Classes of Linear Operators, chapter 0, pages 193-202, Springer.
    • Handle: RePEc:spr:sprchp:978-3-0348-7980-4_5
    DOI: 10.1007/978-3-0348-7980-4_5
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