Wouldn't the "spirit of Jesus" [mentioned in Acts 16:7] challenge the idea of the Trinity?

PROLEGOMENA

Luke, the writer of the words you quoted from Acts 16, had evidently learned well via the Eleven that the Holy Spirit, while not Jesus, was indeed the Spirit of Jesus within the lives of believers after Jesus had ascended to the Father. During Passion Week, Jesus in John 14 had taught his disciples about the person whom he identified by several names and functions, including

• another helper (or comforter), or simply the helper
• the Spirit of truth
• the Holy Spirit
• the one sent by the Father
• the one who would teach Jesus’ followers and bring to remembrance (after Jesus’ ascension) what he had said while with them for three short years

To speak of the Spirit of Jesus, then, is to speak of the Holy Spirit, and vice versa. Jesus’ teaching on the Holy Spirit made one thing perfectly clear: While he—Jesus—would no longer be with his disciples for a time, they could with full assurance know that 1) Jesus would be coming back again, so that where he was, there they would also be (John 14:3); and 2) the Holy Spirit would be, as it were, Jesus’ surrogate presence with them until they were reunited with Jesus in the Father’s house (John 14:2).

Since there is perfect equality within the Godhead, the Bible frequently alludes to this equality in various ways. For example, "I and the Father are one," Jesus said (John 10:30), by which he meant they were of one divine essence. In Acts chapter 5, Peter accused Ananias of lying to the Holy Spirit (v.3), and in the next verse he told Ananias that he had lied to God. Clearly, Peter equated the Holy Spirit with God.

Each person of the trinity is fully God, with all the attributes of deity. I say this notwithstanding that Jesus, on more than one occasion, seemed to defer to the Father, as when he said, for example,

My Father, who hath given [my sheep] unto me, is greater than all; and no one is able to snatch [them] out of the Father's hand (John 10:29 ASV).

On another occasion Jesus also said to his disciples,

You heard that I said to you, “I will go away, and I will come [back] to you.” If you loved me, you would have rejoiced because I go to the Father, for the Father is greater than I (John 14:28).

Jesus' attribution to the Father being "greater than all" does not mean Jesus was or even thought himself to be inferior to the Father in power or in any other way. The point Jesus was making in John 10 was that true believers (i.e., the sheep in Jesus' metaphor) are safely ensconced in the hands of both the Son and the Father. Moreover, both Jesus and the Father are in one accord regarding their plans and purposes for the human race; namely, to call to themselves a people who will be with them forever in heaven and who forever will bear the image of God's only beloved Son, in whom he is well pleased (see Romans 8:29; Matthew 3:17 and 17:5).

According to the Bible, from eternity past, each person in the Godhead has assumed a role, as it were, within the Godhead. Each role is both unique and complementary in the ongoing purposes of all three persons of the Godhead. Moreover, there is unanimity of purpose. Each person will forever be in one accord with the other members of the trinity as to their ongoing, divine motives, intentions, attitudes and actions. Much more could be said about this, but perhaps an illustration might prove helpful.

Just as a baby born into a family begins very early to intuit that each parent is a bit different from the other (if only in their appearance and the sound of their voices), so also in that same family the child will eventually learn that despite the parents having different roles within the family unit, the family is still a unit composed of three equal persons (though very early in life the baby thinks he or she is the center of the universe).

While the father may assume a role which the mother does not assume, the child still intuits that both parents are still parents. At the same time, however, the child relates to each parent in a slightly different way, and vice versa. In a multiple-child family, the same can be said of each child. The parent-child relationship, though it will evolve over time, will always be a human-to-human, parent-child relationship, even if the adult develops dementia and the roles seem to be reversed.

So it is within the Godhead, but on a divine level. Each person—Father, Son, and Holy Spirit, has a role to play, but each person retains his deity and equality within the Godhead. Even when Jesus became a human being via his immaculate and divine conception, his deity was at no time compromised. To be sure, for a time he emptied himself of some of his divine prerogatives, or to put it differently, he veiled his divine nature and attributes, all for the suffering of death (see Philippians 2). At no time, however, did he cease being God.

After Jesus forgave the sins of a paralytic (whom he also healed), even some of Jesus’ detractors were correct when they remarked in Jesus’ presence,

Who is this man who speaks blasphemies? Who can forgive sins, but God alone? (Luke 5:21; Mark 2:7).

Even though Jesus humbled himself by “taking the form of a bond-servant, and [by] being made in the likeness of men,” he still retained the prerogative of God to forgive sins, not to mention perform miracles, which included raising people from the dead! Moreover—and perhaps more importantly, by being the Lamb of God who became “obedient to the point of death, even death on a cross,” Jesus made the forgiveness of sins possible through his shed blood at Calvary (again, see Philippians 2:7-8).

The doctrine of the trinity is perhaps the most difficult-to-understand doctrine in all of Scripture. Christians may sound as if they are copping out when they say it’s a mystery, but it is, in fact, quite mysterious. How can a God who is described as One Lord in one place (e.g., Deuteronomy 6:4) have in another place a “doubting Thomas” say, without being reproved by Jesus for saying it,

”My Lord and my God!” (John 20:28)?

In conclusion, each person in the Godhead, though singularly unique, is in perfect synchronicity with each other person in the Godhead. This fact has been a cornerstone of the Christian faith for centuries, and the teaching was codified in one of Christianity’s most significant creeds, the Athanasian Creed (named after, but likely not written by, Athanasius, the fourth century Bishop of Alexandria), which reads in part,

[W]e worship one God in Trinity, and Trinity in Unity, neither confounding the persons, nor dividing the substance.

For there is one Person of the Father, another of the Son, and another of the Holy Spirit. But the godhead of the Father, of the Son, and of the Holy Spirit, is all one, the glory equal, the majesty co-eternal.

Such as the Father is, such is the Son, and such is the Holy Spirit. The Father uncreated, the Son uncreated, and the Holy Spirit uncreated. The Father incomprehensible, the Son incomprehensible, and the Holy Spirit incomprehensible. The Father eternal, the Son eternal, and the Holy Spirit eternal. And yet they are not three eternals, but one Eternal.

As also there are not three incomprehensibles, nor three uncreated, but one Uncreated, and one Incomprehensible. So likewise the Father is Almighty, the Son Almighty, and the Holy Spirit Almighty. And yet they are not three almighties, but one Almighty.

So the Father is God, the Son is God, and the Holy Spirit is God. And yet they are not three gods, but one God.