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\(L^p\)-nuclearity, traces, and Grothendieck-Lidskii formula on compact Lie groups. (English) Zbl 1296.47019
Authors’ summary: “Given a compact Lie group \(G\), in this paper we give symbolic criteria for operators to be nuclear and \(r\)-nuclear on \(L^p(G)\)-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 noncommutative analogue of the phase space \(G\times\widehat G\), where \(\widehat G\) is the unitary dual of \(G\). No regularity of the kernel (or of the symbol) is assumed so that several of the obtained criteria extend to the more general setting of compact topological groups.”
Applications include one to the trace formula for the heat kernel.

MSC:
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
35S05 Pseudodifferential operators as generalizations of partial differential operators
43A75 Harmonic analysis on specific compact groups
22E30 Analysis on real and complex Lie groups
47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
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