The Mahowald operator in the cohomology of the Steenrod algebra

We study the Mahowald operator $M = langle g_2,h_0^3, - rangle$ in the cohomology of the Steenrod algebra. We show that the operator interacts well with the cohomology of $A(2)$, in both the classical and $mathbb{C}$-motivic contexts. This generalizes previous work of Margolis, Priddy, and Tangora.