Let Gbe a connected compact Lie group. We study the heat operator of a G-transversally elliptic operator. After we review the spectral properties of a transversally elliptic operator, we define a character, that is a distribution on Ggeneralizing the trace of the heat operator  to the G-equivariant case. The main theorem of this paper gives an estimate of a coefficientin the character, which essentially determines the convergence of the character.

transversally elliptic operator, heat operator, distributional character.