Schatten classes and traces on compact groups

作者：Delgado Julio^*; Ruzhansky Michael

来源：Mathematical Research Letters, 2017, 24(4): 979-1003.

DOI：10.4310/mrl.2017.v24.n4.a3

• Compact Lie groups
• topological groups
• pseudodifferential operators
• eigenvalues
• trace formula
• Schatten classes

In this paper we present symbolic criteria for invariant operators on compact topological groups G characterising the Schatten-von Neumann classes S-r(L-2(G)) for all 0 < r = infinity. Since it is known that for pseudo-differential operators criteria in terms of kernels may be less effective (Carleman's example), our criteria are given in terms of the operators' symbols defined on the noncommutative analogue of the phase space G x <(G)over cap>, where G is a compact topological (or Lie) group and (G) over cap is its unitary dual. We also show results concerning general non-invariant operators as well as Schatten properties on Sobolev spaces. A trace formula is derived for operators in the Schatten class S-1(L-2(G)). Examples are given for Bessel potentials associated to sub-Laplacians (sums of squares) on compact Lie groups, as well as for powers of the sub-Laplacian and for other non-elliptic operators on SU(2) similar or equal to S-3 and on SO(3).

• 出版日期2017