Was the spirit of Jesus Christ the Holy Spirit? We say the Jesus is God. Jesus said that God is Spirit. This implies that the Spirit also has a subspirit. The spirit of Jesus was certainly not the Holy Spirit because while dying on the cross he uttered "Father into thy hands I give my spirit." Also when He was resurrected his spirit came back to him and now He is in His corporate body in heaven. This brings me to the conclusion that Jesus Christ had a human spirit and still have the same. How does it fits into Trinity?
There is no reason to separate one's spirit from the person. Read through the lesson on Soul and Spirit.
No, Jesus and the Holy Spirit are not the same being. John 16:7 shows this when Jesus said, "Nevertheless I tell you the truth. It is to your advantage that I go away; for if I do not go away, the Helper will not come to you; but if I depart, I will send Him to you." The Helper, the Holy Spirit, is clearly distinct from Jesus.
Nowhere does the Bible state that Jesus had a human spirit. It states that he came in human form or human flesh, but the body is not the spirit.
For more on the idea of the Godhead and Trinity, read: