There is debate concerning. whether Luke wrote the Gospel of Luke. It can be strongly argued that Luke is in fact the author.
The earliest description of this Gospel that we have says that the Gospel was written by Luke. The book was circulated with Luke in the title.
One argument against Luke being the author is that the titles are not part of the body of the Gospels, so it is suggested that the Gospels were anonymous. However, even if there is no internal evidence that Luke is the author of the Gospel, the unanimous early tradition that ascribes this Gospel to Luke is strong evidence that Luke is indeed its author.
Concerning the four Gospels which alone are uncontroverted in the Church of God under heaven, I have learned by tradition that the Gospel according to Matthew, who was at one time a tax collector and afterwards an Apostle of Jesus Christ, was written first and that he composed it in the Hebrew tongue and published it for the converts from Judaism. The second written was that according to Mark, who wrote it according to the instruction of Peter, who, in his General Epistle, acknowledged him as a son, saying, ‘The church that is in Babylon, chosen together with you, greets you and so does Mark my son.’ And third, was that according to Luke, the Gospel commended by Paul, which he composed for the converts from the Gentiles. Last of all, that according to John.Origin, Commentary on Matthew 1
Here, Irenaeus says that Mark, who is “the interpreter and follower of Peter,” is the author of “his Gospel narrative.” Irenaeus is another early source that says that Mark is the author of the Gospel of Mark.
. . . at which nevertheless he was present, and so he placed. The third book of the Gospel is that according to Luke. Luke, the well-known physician, after the ascension of Christ, when Paul had taken with him as one zealous for the law, composed it in his own name, according to belief. Yet he himself had not seen the Lord in the flesh; and therefore, as he was able to ascertain events, so indeed he begins to tell the story from the birth of John. The fourth of the Gospels is that of John, of the disciples.The Muratorian Fragment
The Muratorian Canon refers to the Gospel of Luke as being the “third” book and the Gospel of John as being the “fourth of the Gospels.”
The Muratorian Canon is another piece of early evidence that Luke wrote the Gospel of Luke.