Schatten class and nuclear pseudo-differential operators on homogeneous spaces of compact groups

Abstract

Given a compact (Hausdorff) group G and a closed subgroup H of G, in this paper we present symbolic criteria for pseudo-differential operators on the compact homogeneous space G/H characterizing the Schatten-von Neumann classes Sr(L 2 (G/H)) for all 0 < r ≤ ∞. We go on to provide a symbolic characterization for r-nuclear, 0 < r ≤ 1, pseudo-differential operators on L p (G/H) with applications to adjoint, product and trace formulae. The criteria here are given in terms of matrix-valued symbols defined on noncommutative analogue of phase space G/H × G/H. Finally, we present an application of aforementioned results in the context of the heat kernels.

L2-Lp estimates and Hilbert–Schmidt pseudo differential operators on the Heisenberg motion group

“…To extend pseudo-differential operators to other settings, one observes that the second R n in the Cartesian product R n × R n is the dual of the additive group R n . These observations allow us to extend the definition of pseudo-differential operators to other groups G, provided we have an explicit formula for the dual of G and an explicit Fourier inversion formula on G. Using this approach, the global theory of pseudodifferential operators on other classes of groups, such as S 1 , Z, affine groups, compact (Lie) groups, homogeneous spaces of compact (Lie) groups, Heisenberg groups, graded Lie groups, step two nilpotent Lie groups, and locally compact type I groups has been widely studied by several researchers [8,15,16,4,21,34,23,6,5].…”

Trace class and Hilbert-Schmidt pseudo differential operators on step two nilpotent Lie groups

“…Over the years, a considerable attention has been devoted by several researchers for finding the criteria for Schatten class of pseudo-differential operators in terms of symbols. Ruzhansky and Delgado investigated this in details in many different settings; for example, using the matrix-valued symbols on compact Lie groups in [7,6,10,9] they successfully characterized these classes of operators on compact Lie groups (see also [19]). Later, they with their collaborators extended these results to compact manifolds and to more general on Hilbert spaces [8,10] using the non-harmonic analysis developed by Ruzhansky and Tokmagambetov [28].…”