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.
Categories: math.FA