Spirit of Holiness is another name of the Holy Spirit. It is likely that it reflects an underlying Hebrew expression for the Holy Spirit that uses the noun (holiness) rather than the adjective (holy).
Spirit of Holiness in Greek:
To pneuma hagiōsynēs
Strong’s Concordance numbers: 4151, 40
Bible reference: Rom. 1:4
Paul began his letter to the Romans by writing about the gospel “promised beforehand through His prophets in the holy Scriptures, concerning His Son, who was born of a descendant of David according to the flesh, who was declared the Son of God with power by the resurrection from the dead, according to the Spirit of holiness, Jesus Christ our Lord.” (Romans 1:2-4)
Notice that Paul states first that Jesus (Yeshua‘) was God’s Son from the start. When Jesus became a man, He was a Son of David. The resurrection confirmed the truth that Jesus is the Son of God and that He is the Messiah (Christ), worthy of the title “Lord.”