Characterizations of self-adjointness, normality of pseudo-differential operators on homogeneous space of compact groups

Abstract

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.

Keywords:

• Pseudo-differential operators
• homogeneous spaces of compact groups
• self-adjoints
• normal
• eigenvalues
• eigenfunctions

AMS Subject Classifications:

• Primary 47F05
• 47G30
• Secondary 43A85

Acknowledgments

Shyam Swarup Mondal thanks IIT Guwahati for providing financial support.

Disclosure statement

No potential conflict of interest was reported by the author(s).