Kernel and symbol criteria for Schatten classes and $r$-nuclearity on compact manifolds

Abstract

In this note we present criteria on both symbols and integral kernels ensuring that the corresponding operators on compact manifolds belong to Schatten classes. A specific test for nuclearity is established as well as the corresponding trace formulae. In the special case of compact Lie groups, kernel criteria in terms of (locally and globally) hypoelliptic operators are also given. A notion of an invariant operator and its full symbol associated to an elliptic operator are introduced. Some applications to the study of $r$-nuclearity on $L^p$ spaces are also obtained.

arXiv comment8 pages. arXiv admin note: substantial text overlap with arXiv:1404.6479

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• Lp-Nuclearity, traces, and Grothendieck-Lidskii formula on compact Lie groups
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Cited by in corpus (11)

• $L^p$-boundedness and $L^p$-nuclearity of multilinear pseudo-differential operators on ${\mathbb{Z}}^n$ and the torus ${\mathbb{T}}^n$
• Global Hypoellipticity for First-Order Operators on Closed Smooth Manifolds
• Global Hypoellipticity for Strongly Invariant Operators
• On some spectral properties of pseudo-differential operators on T
• Spectral properties of pseudo-differential operators over the compact group of $p$-adic integers and compact Vilenkin groups
• On the nuclear trace of Fourier integral operators in $L^p$-spaces
• Fourier multipliers in Hilbert spaces
• Nuclearity for Fourier integral operators in $L^p$-spaces
• Geometric maximizers of Schatten norms of some convolution type integral operators
• Nuclear pseudo-differential operators in Besov spaces on compact Lie groups
• Schatten properties, nuclearity and minimality of shift invariant spaces