Schatten classes and traces on compact groups
Schatten classes and traces on compact groups
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DOI：
10.4310/mrl.2017.v24.n4.a3
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发表时间：
2013-03
期刊：
影响因子：
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通讯作者：
J. Delgado;Michael Ruzhansky
中科院分区：
文献类型：
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作者：
J. Delgado;Michael Ruzhansky
本文给出了紧拓扑群G上刻画Schatten-von Neumann类S_{r}（L^{2}（G））的不变算子的符号判据，其中0 <r\leq\infty。由于众所周知，对于伪微分算子，根据核的标准可能不太有效（Carleman的例子），我们的标准是根据定义在相空间$G\times\hat{G}$的非交换模拟上的算子符号给出的，其中$G$是紧致拓扑（或李）群，$\hat{G}$是它的酉对偶。我们还展示了关于一般非不变算子以及Sobolev空间上的Schatten性质的结果。给出了Schatten类S_{1}（L^{2}（G））中算子的迹公式.例子给出了与紧李群上的次拉普拉斯算子（平方和）相关的贝塞尔势，以及次拉普拉斯算子的幂和SU（2）$\simeq\mathbb S^3$和SO（3）上的其他非椭圆算子。
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 $0<r\leq\infty$. 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 $G\times\hat{G}$, where $G$ is a compact topological (or Lie) group and $\hat{G}$ 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)$\simeq\mathbb S^3$ and on SO(3).