Lp-Nuclearity, traces, and Grothendieck-Lidskii formula on compact Lie groups

2013

59 citations recorded by the source corpus.

Given a compact Lie group $G$, in this paper we give symbolic criteria for operators to be nuclear and r-nuclear on Lp-spaces, with applications to distribution of eigenvalues and trace formulae. Since criteria in terms of kernels are often not effective in view of Carleman's example, in this paper we adopt the symbolic point of view. The criteria here are given in terms of the concept of matrix symbols defined on the non-commutative analogue of the phase space $G\times\hat{G}$, where $\hat{G}$ is the unitary dual of $G$.

View Paper: https://www.semanticscholar.org/paper/66a56b2d9a95a49b597ec3b4795e29d0382b0816

Citations in the corpus

• Global functional calculus for operators on compact Lie groups (2013, unknown): “ norm of σand varying the powers of dξ, for which we refer to [RT10, Section 10.3.3]. For a discussion on the relation between the spaces ℓp(Gb) and Schatten classes of operators on L2(G) we refer to [DR13b]. As Hξ are minimal bi-invariant subspaces of L2(G), they are eigenspaces of all biinvariant ”
• Kernel and symbol criteria for Schatten classes and r-nuclearity on compact manifolds (2014, unknown): “endieck [Gro55] for invariant operators in terms of the full matrix-symbol. The problem of finding such criteria on different kinds of domain is classical and has been much studied (cf. [BS77], [DR13], [DR14b], [Tof08], [BT10] and references therein). Date: July 2, 2018. The first author was supported ”
• Fourier multipliers, symbols, and nuclearity on compact manifolds (2014, unknown): “ the nuclearity of operators inLp(M) We now turn to the study of nuclearity in Lp-spaces on closed manifolds. Sufficient conditions for r-nuclearity on Lp on compact Lie groups have been established in [DR14a]. The study of nuclearity on Lpin this section relies on the analysis of suitable kernel deco”
• Schatten classes, nuclearity and nonharmonic analysis on compact manifolds with boundary (2015, unknown): “d Tr(T) is as in (1.6). We will refer to (1.7) as the Grothendieck-Lidskii formula. Grothendieck established applications to the distribution of eigenvalues of operators in Banach spaces. We refer to [DR14b] for several conclusions in the setting of compact Lie groups concerning summability and dist”
• Schatten classes and traces on compact groups (2013, unknown): “ppearing non-trivial multiplicities of the eigenvalues of the Laplacian on the noncommutative compact Lie groups; for 0 < r ≤ 1, the r-nuclearity on Lp-spaces on compact Lie groups is addressed in [DR14d]. In order to illustrate the relevance of our symbolic criteria we briefly recall a classical ”
• Schatten classes on compact manifolds: Kernel conditions☆ (2014, unknown): “in [Tof06], [Tof08], [BT10]. Schatten classes on compact Lie groups and s-nuclear operators on Lp spaces from the point of view of symbols have been respectively studied by the authors in [DR13b] and [DR13a]. In Date: March 26, 2014. 1991 Mathematics Subject Classification. Primary 47G10; Secondary 4”