Fourier multipliers, symbols, and nuclearity on compact manifolds

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...

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Bibliographic Details
Published inJournal d'analyse mathématique (Jerusalem) Vol. 135; no. 2; pp. 757 - 800
Main Authors Delgado, Julio, Ruzhansky, Michael
Format Journal Article
LanguageEnglish
Published Jerusalem The Hebrew University Magnes Press 01.06.2018
Springer Nature B.V
Subjects
Abstract Harmonic Analysis
Analysis
Dynamical Systems and Ergodic Theory
Fourier analysis
Functional Analysis
Hilbert space
Invariants
Mathematics
Mathematics and Statistics
Multipliers
Operators
Partial Differential Equations
Subspaces
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Summary: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 L 2 ( M ) and to obtain a formula for the trace of trace class operators. We also apply it to provide conditions for operators between L p -spaces to be r -nuclear in the sense of Grothendieck.
ISSN:0021-7670
1565-8538
DOI:10.1007/s11854-018-0052-9