If one takes The Bible seriously, in short, "Yes".
Any time there is a problem of understanding who or what particular parts of Scripture refer to, put 'Christ' and/or 'Messiah' and see what happens. However, one must also use common-sense, and not base interpretation solely on only particular reference but cross-reference it with others that also back it up.
For example
(Isaiah 53 Verse 3) see Matthew 27:30, 31
(Verse 4) See Matthew 8:17, 1Peter 2;24
(Verse 7) See matthew 26:62, 63
(Verse 9) See Matthew 26:57-60
Another answer
Isaiah 53 is personified "becomes a person" in Christ. It was a description at the mouth of the prophet of Christ, his purpose and his life. Most people do not understand the reason for them being on this earth at this time. Most do not understand that their time here is probation related to the events which occurred in the 1st earth age (2nd Peter chapter 3).
Hebrews 10:7 "Then said I, Lo I come in the volume of the book its is written of men to do they will O God" Further reading of Hebrews 10 will match Isaiah 53 prophecy to its occurrence when Christ was born and walked the earth as flesh man, as the sacrifice for one and ll time for our sins.
First was the prophecy given Second was the fulfillment of the prophecy
This method gives Christians a solid foundation to know that the word of our Father is complete and accurate in every detail. We are observing, experiencing the occurrence of prophecy today as it is defined in Matthew 24, Mark 13, and Luke 21.
Isaiah 53 is part of the writings of Second Isaiah (mainly Isaiah chapters 40-55), written during the Babylonian Exile, originally and separate from the work by Isaiah, son of Amoz. We can see clearly from chapter 52 that this was written at a time when the Jews were soon to return to Jerusalem, especially verse 52:9: "Break forth into joy, sing together ye waste places of Jerusalem: for the lord hath comforted his people, he hath redeemeth his people."
Second Isaiah often used the imagery of the suffering servant. We see the 'servant' in 52:13, and chapter 53 is arguably a reference throughout to this servant. In some cases, the 'servant' seems to refer to a particular person, sometimes probably even himself, and sometimes the nation of Judah. In this case, the servant is probably the nation of Judah - we see (53:2) that he [the servant] will grow up before him [God] as a tender plant ... (53:7) that he was oppressed; and so on.
According to Jewish tradition, in Isaiah 53 the entire Jewish nation is referred to in the singular (Rashi commentary, 53:3). The chapter is speaking of the persecutions against the Jews during the last 2000 years.See also:
Isaiah 51:17
Isaiah 53:2
Read Isaiah chapter 53.
The Romans were told and showed that he was the son of God, but chose not to believe it. Also, this was apart of God's plan. Throughout the first testament God slowly reveals his plan. Isaiah 7:14, Isaiah 8:6-8, Isaiah 11:10-12, Isaiah 16:5, Isaiah 53:3-6, and Isaiah 53:11.
Isaiah 53:7 and Acts 8:32
Isaiah 53:3 - This is referring to Jesus
Isaiah 53:4,5 Matt.8:17
Because of Jesus' sacrificial role, John the Baptizer identified Jesus as "the Lamb of God that takes away the sin of the world. In speaking of Jesus as a Lamb, John may have had in mind the words of Isaiah: "He was being brought just like a lamb to the slaughtering." (Isaiah 53:7) "He poured out his soul to the very death," prophesied Isaiah. (Isaiah 53:12)
God foretold this via His prophets: See Isaiah 52:13 thru Isaiah 53 as an example.
The prophet Isaiah, was to prophecy, the birth of christ 700 years before the birth of Christ. He also in chapter 53 clearly tells of Christs trial and punishment for us.
Isaiah 53 portrays the suffering and death of Christ, often referred to as the "Suffering Servant" passage. It describes the sacrificial nature of Jesus' death and how he would bear the sins of many. This chapter is significant in highlighting the atoning work of Christ on the cross.
The prophecy that salvation would come through the suffering servant of the Lord is found in the Book of Isaiah in the Hebrew Bible (Old Testament). Isaiah himself is credited with stating this prophecy, specifically in Isaiah 53.