Split dual Dyer-Lashof operations (Q1295531)

Let \(X\) be a connected space and \(I=(r_1,r_2,\dots,r_s)\) be an admissible sequence. The author proves that there exist functions \(\widehat Q_I:T\overline H(X)\to H^*(\Omega^n\Sigma^nX)\) which are ``dual'' to the corresponding Dyer-Lashof operations \(Q_I\) and such that the (not direct) sum \(\sum_I\text{Im} \widehat Q_I\), taken over all admissible \(I\), contains a generating system for \(H^*(\Omega^n\Sigma^nX)\).