A website to help you study the Gospel of Luke,

The style is unmistakeably similar.
Both books are dedicated to Theophilus.
Acts mentions a `former treatise' by the same author, covering the life of Jesus.
The same concerns are in both books: they portray the Christian church as the outcome of God's promised activity in history, centring on Jesus, witnessed and attested by the experience of real living people.
Some of the phrases are distinctive and similar. `Simon the Zealot' is described in this way only in Luke and in Acts. The Holy Spirit is described in both books as `the promise of the Father'. The insistence upon the `witness' of the apostles is there in both books, much more than in any of the other gospels.
Luke's name is associated with the Gospel from the earliest lists (such as the Muratorian Fragment). He's claimed as an Evangelist by Irenaeus, just a century later; he also quotes parts of the Gospel and ascribes them to Luke. Tertullian and Clement of Alexandria also assume that the Gospel was written by Luke; and there's no competing claim anywhere else.

The Muratorian Fragment gives us this information:

The third Gospel book, that according to Luke. This physician Luke after Christ's ascension, since Paul had taken him with him as an expert in The Way, composed it in his own name according to his thinking. Yet neither did he himself see the Lord in the flesh; and therefore, as he was able to discern it, so he begins to tell the story from the birth of John.

Although the Gospel was designed simply for Theophilus, it was recognized right from the start as inspired and canonical. Justin Martyr obviously used it (he quotes many details which are found only in Luke) and Tatian lifted chunks to insert in the Diatessaron. Amongst heretics, Marcion and the Valentinians used it heavily.

There's a slightly different version of Luke called the `Western text', probably edited by someone unknown around AD 120. It seems that the reading of 20:13 in this version was: `But I Luke, and those who were with me, went on board'. Even if it wasn't Luke who produced this reading (and it probably wasn't!), it does mean that by 120 AD people already assumed that Luke was the author. That's within about half a century of the first writing of the book.

What's more, any book which had a dedication (such as the one to Theophilus) would also have the name of its author attached to it, probably on a little tag. So it seems unlikely that anybody else could have written the Gospel, with his name attached to it... and then be completely forgotten about within half a century!