Fourier multipliers, symbols, and nuclearity on compact manifolds
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The notion of invariant operators, or Fourier multipliers, is discussed for densely defined operators on Hilbert spaces, with respect to a fixed partition of the space into a direct sum of finite-dimensional subspaces. As a consequence, given a compact manifold M endowed with a positive measure, we introduce a notion of the operator’s full symbol adapted to the Fourier analysis relative to a fixed elliptic operator E. We give a description of Fourier multipliers, or of operators invariant relative to E. We apply these concepts to study Schatten classes of operators on L2(M) and to obtain a formula for the trace of trace class operators. We also apply it to provide conditions for operators between Lp-spaces to be r-nuclear in the sense of Grothendieck.
- [Ati68]M. F. Atiyah, Global aspects of the theory of elliptic differential operators , in Proc. Internat. Congr. Math. (Moscow, 1966), Izdat. “Mir”, Moscow, 1968, pp. 57–64.Google Scholar
- [Dix77]J. Dixmier, C*-algebras. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.Google Scholar
- [Dix96]J. Dixmier, Les algèbres d’opérateurs dans l’espace hilbertien (algèbres de von Neumann), Editions Jacques Gabay, Paris, 1996. Reprint of the second (1969) edition.Google Scholar
- [Kön78]H. König, Eigenvalues of p-nuclear operators, in Proceedings of the International Conference on Operator Algebras, Ideals, and their Applications in Theoretical Physics (Leipzig, 1977), Teubner, Leipzig, 1978, pp. 106–113.Google Scholar
- [See67]R. T. Seeley. Complex powers of an elliptic operator, in Singular Integrals, Amer. Math. Soc., Providence, RI, 1967, pp. 288–307.Google Scholar
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