  Let $p\in[1,\infty]$. Being motivated by weakly $p$-convergent and
weak$^\ast$ $p$-convergent operators between Banach spaces introduced by Fourie
and Zeekoei, we introduce and study the classes of weakly $p$-convergent and
weak$^\ast$ $p$-convergent operators between arbitrary locally convex spaces.
Relationships between these classes of operators are given, and we show that
they have ideal properties. Numerous characterizations of weakly $p$-convergent
and weak$^\ast$ $p$-convergent operators are given.
