While the Holy Spirit is the active presence of God in the world and the church, the Old Testament uses names that identify the Holy Spirit with God.

The Spirit of God

The earth was formless and void, and darkness was over the surface of the deep, and the Spirit of God was moving over the surface of the waters. (Gen 1:2)

The Spirit of the Lord

The Spirit of the Lord came upon him, and he judged Israel. When he went out to war, the Lord gave Cushan-rishathaim king of Mesopotamia into his hand, so that he prevailed over Cushan-rishathaim. (Jud 3:10)

Here in this account, God's covenant name, Yĕhovah, is used.

The Spirit of the Lord God

Jesus uses names for the Holy Spirit that reflect His Deity and something about His relationship to human beings.

The Spirit of Your Father

For it is not you who speak, but it is the Spirit of your Father who speaks in you. (Matt 10:20)

The Promise of My Father

And behold, I am sending forth the promise of My Father upon you; but you are to stay in the city until you are clothed with power from on high. (Luke 24:49)

In this instance, Jesus is making a reference to the New Covenant.

The apostles use names for the Holy Spirit that express His Deity and as part of God's triune nature.

The Spirit of the Lord

When they came up out of the water, the Spirit of the Lord snatched Philip away; and the eunuch no longer saw him, but went on his way rejoicing. (Acts 8:39)

The Spirit of Him Who Raised Jesus from the Dead

But if the Spirit of Him who raised Jesus from the dead dwells in you, He who raised Christ Jesus from the dead will also give life to your mortal bodies through His Spirit who dwells in you. (Rom 8:11)

Spirit of Our God

Such were some of you; but you were washed, but you were sanctified, but you were justified in the name of the Lord Jesus Christ and in the Spirit of our God. (1 Cor 6:11)

Spirit of the Living God

You are our letter, written in our hearts, known and read by all men; being manifested that you are a letter of Christ, cared for by us, written not with ink but with the Spirit of the living God, not on tablets of stone but on tablets of human hearts. (2 Cor 3:2-3)

The Spirit of His Son

Because you are sons, God has sent forth the Spirit of His Son into our hearts, crying, "Abba! Father!" (Gal 4:6)

The Spirit of Jesus Christ

What then? Only that in every way, whether in pretense or in truth, Christ is proclaimed; and in this I rejoice. Yes, and I will rejoice, for I know that this will turn out for my deliverance through your prayers and the provision of the Spirit of Jesus Christ, according to my earnest expectation and hope, that I will not be put to shame in anything, but that with all boldness, Christ will even now, as always, be exalted in my body, whether by life or by death. (Phil 1:18-20)