On the first day of sukkos there is a din of “lechem” by the arba minim that they must belong to you in order to fulfill the mitzvah. After someone has already made a bracha on his arba minim and fulfilled the mitzvah when the person then takes the arba minim again to shake them by Hallel must they also still “belong” to him with all of the dininm of “lechem”. Or may he use someone else’s (without making some sort of kinyan (acquisition) on them.)
Shaking the lulav during Hallel does not require “lachem” even on the first day of Sukkos.
The halachah of “lachem” is stated about the Torah mitzvah of taking the lulav on the first day, and even for taking the lulav on the remaining days of Sukkos (outside the mikdash) there is no obligation of “lachem.” It stands to reason that there is likewise no obligation of “lachem” for shaking the lulav during Hallel.
This reasoning is mentioned by a number of acharonim.