I’ll start off the debate on trinities in various religions with a very simple (sceptical) observation:
I very much doubt it makes sense to pretend the various trinities in various religions can be equated. Father-Son-Holy Spirit is the Christian trinity and it says something about the relationship between the divine and each individual, as well as about how God relates to man. The son is said to have died for us, while the Father stayed immaculate, or something. The Holy Spirit is the one way in which God can communicate with normal human beings like us – and I would say it might (with a stretch) be equated with our Buddhi: that which mediates between the divine (Atman) and our ordinary personalities (kama-manas).
Theosophist 