An important hermeneutical tool is to allow scripture to interpret scripture. What do these parallel passages tell us about the term “Holy Spirit”?
for the Holy Spirit will teach you in that very hour what you ought to say." Luke 12:12
for it will not be you speaking, but the Spirit of your Father speaking through you. Matthew 10:20.
Holy Spirit is the Spirit of the Father because He (Spirit) issues from Him (Father) (John 15:26), and is also the Spirit of the Son, because He (Spirit) is dispensed by Him (the Son) to humans (John 20:22) and because belief in Him (Son) is the condition for Holy Spirit to gush forth in us as streams of "living water" (John 7::38).
Thus, yes, according to apostles and according to common sense based on their sayings and writings, the Holy Spirit is the Spirit of the Father who teaches us to worship the Father's Son, called Jesus Christ after the Incarnation, alongside with the Father as Lord and God (1 Cor. 12:3).
This question is perhaps more suitable for Christianity.SE, as it is asking for answers that could depend on specific faiths.
From a strictly textual perspective, "the holy spirit" and "the spirit of the Father" mean the same thing. What belongs to or comes from the Father can be considered as holy, so obviously spirit from or of the Father must be holy.
The problem of reconciling these two verses arises only when a non-biblical[*] concept is added, namely that holy spirit is a third person in the godhead.
Ignoring later tradition, and working only with biblical text (which is what this site is supposed to do), there is no reason to even suspect that there is any conflict between the two verses.
[*] before anyone jumps on this, remember that in the original Greek, "holy spirit" wasn't capitalized, English translations referring to it as "he" could have used the word "it" instead, and the Johannine Comma is a late addition.
The Holy Spirit is known by various titles:
The Holy Spirit is seen as distinct from both the Father and the Son in many places such as: Matt 28:19, 2 Cor 12:13, Eph 2:18, John 16:13, Acts 10:38, Rom 1:2-4, 1 Cor 6:11, 12:2, 3, 2 Cor 13:14, Gal 3:11-14, Heb 9:14, 10:29, 1 Peter 1:2, 3, 4:14, etc.
We see the individuality of the Holy Spirit from the Father and Son in other passages such as Matt 12:31, 32, Mark 3:28, 29, and Luke 12:8-10 the unforgivable sin is defined as blasphemy against the Holy Spirit. This is an expansion of Isa 63:10-14 where people grieved the Holy Spirit. Such a sin would not be even possible if the Holy Spirit were not both a person and divine. Note further, that these passages make a clear distinction between sinning against the Son or Father as opposed to the Holy Spirit, again, shown that the Holy Spirit is distinct.
Rom 8:26, 27, which says – In the same way, the Spirit helps us in our weakness. For we do not know how we ought to pray, but the Spirit Himself intercedes for us with groans too deep for words. And He who searches our hearts knows the mind of the Spirit, because the Spirit intercedes for the saints according to the will of God.
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Now, there is no question that "the spirit of Christ" can, sometimes, refer to the mind of Christ - see BDAG, "pneuma", meaning #6.
The word, "Holy" comes from the Hebrew word, qodesh (Strong's H6944), which means apartness, set-apartness, of God, of places, of things, also, set-apartness, separateness, sacredness. So the Holy Spirit is indeed of God, especially when viewed in His separate operative capacity, like "comforter", "teacher", one by whom we are "sealed", etc.
This is not unlike the separate operative capacity of the WORD of God--Spirit--who was made flesh to be judged for the sin of the world by being separated from God the Father in Hell for three days and three nights as prophesied in Psalm 22, And, as fulfilled in Matthew 27:46:
And about the ninth hour Jesus cried with a loud voice, saying, Eli, Eli, lama sabachthani? that is to say, My God, my God, why hast thou forsaken me?
and finishing with the same cry, in verse 50:
Jesus, when he had cried again with a loud voice, yielded up the ghost. (My emphasis)
And, in Acts 2:25-27:
For David speaketh concerning him, I foresaw the Lord always before my face, for he is on my right hand, that I should not be moved: Therefore did my heart rejoice, and my tongue was glad; moreover also my flesh shall rest in hope: Because thou wilt not leave my soul in hell, neither wilt thou suffer thine Holy One to see corruption. (My emphasis)
David was consoled concerning his own soul because of his faith in the promise that God would not suffer His Holy One to see corruption, neither would He leave David's own soul in Hell. As it was written, in Psalm 16:10:
For thou wilt not leave my soul in hell; neither wilt thou suffer thine Holy One to see corruption. (My emphasis)
And again in Matthew 12:40:
For as Jonas was three days and three nights in the whale's belly; so shall the Son of man be three days and three nights in the heart of the earth.
The heart of the earth was not in that tomb where Jesus' body remained, but rather, in the heart of the earth--in hell--in Abraham's bosom--separated by a great gulf fixed from unbelievers (read Luke, chapter 16 in total). Those believers in Abraham's bosom in hell at that time, although separated as being believers from unbelievers, were not in heaven. They, too, were in the heart of the earth separated from their own long-ago decayed bodies, because Jesus had not yet completed the necessary work for their salvation. Indeed, He was now separated from the Father, just like them, but now positioned to preach to ALL those spirits in prison words that had they been waiting years, decades, and even thousands of years to hear--that God's Holy One had finally died as an atonement for their sin, as it is written in 1 Peter 3:18-19:
For Christ also hath once suffered for sins, the just for the unjust, that he might bring us to God, being put to death in the flesh, but quickened by the Spirit: By which (Spirit) also he went and preached unto the spirits in prison; (My insertion for clarification)
That Word of God revealed the separate operational capacities of Himself, the Speaker, the Lord God, and the His (the Lord God's) Spirit, the Senders, in Isaiah 48:16:
Come ye near unto me, hear ye this; I have not spoken in secret from the beginning; from the time that it was, there am I: and now the Lord GOD, and his Spirit, hath sent me. (My emphasis)
These three separate operational capacities of the ONE TRUE GOD indeed have individual functions and they always stayed in their own lane, respectively, while accomplishing ETERNAL SALVATION for everyone who believes in the separate, but dovetailing accomplishments of the Father, the Son, and the Holy Ghost. Those accomplishments of the Holy Father, Holy Son, and Holy Ghost are brilliantly highlighted as basic necessities for teaching by the great commission given to the apostles, who were likewise "sent" to "teach" all nations by baptizing (submerging) them in the name of the three individual operatives of the ONE True God, as we see in Matthew 28:19-20:
Go ye therefore, and teach all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Ghost: Teaching them to observe all things whatsoever I have commanded you: and, lo, I am with you alway, even unto the end of the world. Amen. (My emphasis)
So the Holy Spirit is indeed God, Even as the WORD who was with God and who was God, is God, and as the Lord God is God--the ONE TRUE GOD--the "us" in Genesis 1:26:
And God said, Let us make man in our image, after our likeness:.... (My emphasis)
So, yes, the Holy Spirit is of the Father.
for the Holy Spirit will teach you in that very hour what you ought to say." Luke 12:12
for it will not be you speaking, but the Spirit of your Father speaking through you. Matthew 10:20
Perhaps these passages don’t fit the prescribed understanding of what (or who) the Holy Spirit is, but they carry the weight of those Jesus ministered to as his disciples.
The disciples, become apostles, carry the weight of holding, guarding and teaching truth once delivered.
What they present is simple - that the Holy Spirit IS the spirit of the Father.
There is no separate entity. There is no separate entity when we read of the spirit of Christ, or the several others so described. (Elijah, truth, grace, fire, etc)
Jesus, full of the Holy Spirit, returned from the Jordan and was led around by the Spirit in the wilderness Luke 4:1
Which we may understand as him being filled with the spirit of his Father. Making perfect sense of Jesus describing he and ‘the Father were one’. Not one substance, but one in purpose. Jesus fully equipped to deal with evil without succumbing - not by himself, but filled with the Father - the spirit of God, the spirit of the Father.
For God was pleased to have all His fullness dwell in Him Col 1:19
