Let G be a compact Hausdorff group and H be a closed subgroup of G. In this paper, we show that every bounded linear operator T on $L^p(G/H)$ is a pseudo-differential operator with the symbol σ for $1\le p<\mathrm{\infty }$. We present necessary and sufficient conditions on symbols to ensure that a bounded pseudo-differential operator on $L^2(G/H)$ is self-adjoint, normal and present explicit formula for their symbols. A necessary and sufficient condition is also given such that the bounded linear operators on $L^p(G/H)$ posses eigenvalues and eigenfunctions.

中文翻译：

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紧群齐次空间上伪微分算子的自伴随性、正态性表征

令G为紧 Hausdorff 群，H为G的闭子群。在本文中，我们证明了每个有界线性算子T${\mathrm{大号}}^p(G/H)$是一个伪微分算子，符号σ为$1\le p<\mathrm{\infty }$. 我们提出了关于符号的充分必要条件，以确保有界伪微分算子在${\mathrm{大号}}^2(G/H)$是它们符号的自伴随、正常和现在的显式公式。还给出了一个充分必要条件，使得有界线性算子${\mathrm{大号}}^p(G/H)$具有特征值和特征函数。