25 Now I think that it ought to be clearly understood that God the Father is denoted by the Spirit of God, because our Lord Jesus Christ declared that the Spirit of the Lord was upon Him since He anoints Him and sends Him to preach the Gospel. For in Him is made manifest the excellence of the Father's nature, disclosing that the Son partakes of His nature even when born in the flesh through the mystery of this spiritual unction, since after the birth ratified in His baptism this intimation of His inherent Sonship was heard as a voice bore witness from Heaven:— You are My Son; this day have I begotten You. For not even He Himself can be understood as resting upon Himself or coming to Himself from Heaven, or as bestowing on Himself the title of Son: but all this demonstration was for our faith, in order that under the mystery of a complete and true birth we should recognise that the unity of the nature dwells in the Son Who had begun to be also man.
We have thus found that in the Spirit of God the Father is designated; but we understand that the Son is indicated in the same way, when He says: But if I in the Spirit of God cast out devils, then has the kingdom of God come upon you. That is, He shows clearly that He, by the power of His nature, casts out devils, which cannot be cast out save by the Spirit of God. The phrase 'Spirit of God' denotes also the Paraclete Spirit, and that not only on the testimony of prophets but also of apostles, when it is said:— This is that which was spoken through the Prophet, It shall come to pass on the last day, says the Lord, I will pour out of My Spirit upon all flesh, and their sons and their daughters shall prophesy. And we learn that all this prophecy was fulfilled in the case of the Apostles, when, after the sending of the Holy Spirit, they all spoke with the tongues of the Gentiles.
Source: On the Trinity (New Advent)