You've already seen that in Italian, after expressions related to opinions, feelings, preferences, or desires, you will need to use verbs in the subjunctive. For example, Spero che lui trovi qualcosa.
But if the second verb (trovare) refers to something that happened before the first verb (Spero), you'll have to use a different subjunctive form.
Spero che tu abbia trovato qualcosa.
I hope you found something.
This special verb form is the combination of the subjunctive of avere plus a verb that ends in ‑ato, ‑uto, or ‑ito.
Lui pensa che io l'abbia lasciato.
He thinks I left him.
Credo che lei abbia capito.
I believe that she has understood.
Like other two‑part verbs in Italian, you use a form of avere a lot of the time, but when you want to talk about motion, you need essere. You might remember using it with verbs like venire and andare.