Let <i>G</i> be any compact connected Lie group and let <i>T</i> ≤ <i>G</i> be a maximal torus.  Then for any unstable Adams operation <i>f</i> of degree <i>k</i>, the following diagram commutes up to homotopy «!» And conversely, any map <i>f</i> that makes the above diagram commute must be an unstable Adams operation. Using this characterization, we will construct a self-map of a <i>p</i>-local compact group (<i>S,F,L</i>) in order to define unstable Adams operations on a more general setting. THEOREM.  For any <i>p</i>-local compact group (<i>S,F,L</i>) there is a self-equivalence such that the map on the objects when restricted to the identity component of <i>S</i> is a <i>q^m-th </i>power map.