  In this paper we develop the functional calculus for elliptic operators on
compact Lie groups without the assumption that the operator is a classical
pseudo-differential operator. Consequently, we provide a symbolic descriptions
of complex powers of such operators. As an application, we give a constructive
symbolic proof of the G\r{a}rding inequality for operators in
$(\rho,\delta)$-classes in the setting of compact Lie groups.
