What are the specific tasks of each of the elements (that is, each of the three “persons”) of the Holy Trinity, according to the Christian faith?

When the one God is love (mutual “self-giving”, see 1 Jn 4:7-21), then the three “persons” in one are simultaneously the “intersections” between which the rhythm of love is consummated: Giving – Receiving – Returning (Please note that “Person” in this context has a different significance to that of a person with a human personality as an “independent” in itself-centred reality). With that, all three “Persons” are one and the same love in three beings that are indispensable so that God can even be love at all, and – moreover… the highest selfless love. The one God is community, which means, he is the one loving game that takes place between the three “Persons”: Love, to be loved, to love with others.
In 1 Jn 1:3 it says: “We declare to you what we have seen and heard so that you also may have the fellowship with us; and truly your fellowship with the Father and with his Son Jesus Christ.“

Everything points to unity. And here, also, it is not about a “commonplace unity”, but a unity that receives its model from the three-in-one God. It is a unity that is realised in its variety and a variety that is moving towards unity. It is communio (communion, community).

The unity that the whole world strives for, the desire for unanimity, harmony and peace that is in every one of us, yes, the “globalisation”, interconnection and universal communication, which all technology, media and culture aim for, have something in common with the Holy Trinity, with Christian belief in the one God in three “Persons.” More precisely: There should be equivalence, a similarity between the two. This confirms this thesis once more: What God is, communio, is what we can and

must become. He is the beginning of all and the target of all reality. In him, we are, we live and we move in him.
When God works on us, he always work as the one, three-in-one, God.