Can follower take items all on his/her own?

I gave my follower (Lydia) basic items to improve her chances to survive: armor, weapon, shield, bow and necklace. I also gave her some items to keep if I ever need them for immediate use e.g. fire resistance boots.

However, when I check her inventory sometimes new items appear apparently out of nowhere: usually all kinds of arrows but sometimes swords and last night enchanted stuff to turn undead.

1. I'm 100% sure I never gave her those items.
2. I'm 100% sure I never gave her command to pick those items up from the floor or from a container.
3. The arrows might be explained if she picks up arrows shot at her - she take lots of those very bravely without ever complaining but so far wasn't able to see her taking arrow out her body and put in her quill.
4. I do explore lots of places that most likely have the items appearing in her inventory.

Can I control this thing or define what she should take and what not to take? Is this behavior documented somewhere? If so, I couldn't find it so far.

Solution 1:

Arrows have a chance of being added to the inventory of an NPC or creature they hit. There is no animation for pulling them out, they just instantly end up in their inventory.

NPCs without a weapon in their inventory will go after weapons laying on the ground and in containers, perhaps that's the reason for melee weapons in your companion's inventory. They might also go for unattended staves, though I've never observed such behavior.