Were the Gospel of Luke and the book of Acts written by the same person? Can you please explain why some people would argue that the Gospel of Luke and Acts were written by the same person?
The fact that both Books were written by the same individual known as Luke is extracted from the first few verses of both Books.

Luke 1:1 through4 KJV Forasmuch as many have taken in hand to set forth in order a declaration of those things which are most surely believed among us, Even as they delivered them unto us, which from the beginning were eyewitnesses, and ministers of the word; It seemed good to me also, having had perfect understanding of all things from the very first, to write unto thee in order, most excellent Theophilus, That thou mightest know the certainty of those things, wherein thou hast been instructed.
 Acts 1:1 KJV , The former treatise have I made O Theophilus, of all that Jesus began both to do and teach,

Both books were written to one Theophilus, and  The former treatise have I made  O Theophilus indicates that Acts is a follow-up to the Book of Luke.
These two commentary excerpts deal with these introductions to the books.

&quot;Luke 1:1
Forasmuch as many have taken in hand,.... From hence, to the end of Luk_1:4 is a preface of the evangelist to his Gospel, setting forth the reasons of his writing it; and which he wrote and sent to the excellent Theophilus, for the further confirmation of him in the faith of Christ. It seems that many had took in hand, or attempteo set forth in order a declaration of those things which are most surely believed among us; that is, they undertook to write and publish a very particular and exact narrative of the birth, life, actions, doctrines, miracles, sufferings, death, resurrection, and ascension of Jesus Christ; things which Luke, and other Christians, had the fullest and strongest evidence, and were confidently assured of, and most firmly believed, even with a full assurance of faith. By these many, he cannot mean the authentic historians of evangelical facts, as Matthew and Mark; for they two cannot, with any propriety, be called many; and besides, it is not so very clear and certain a point, that they had, as yet, wrote their Gospels; nor would this evangelist suggest any deficiency, weakness, and inaccuracy in them, as he seems to do: nor does he intend such spurious writers as the authors of the Gospels according to the Nazarenes, Hebrews, and Egyptians; of Nicodemus, Thomas, Matthias, and of the twelve apostles; and still less, the Gospels of Cerinthus, Basilides, and other heretics; since these would not have passed without a censure from him, for the falsehood, fabulous, and trifling stuff in them, as well as for the wicked and heretical opinions propagated by them; and besides, these pieces were not extant when this Gospel was written: but he seems to design some honest and well meaning Christians, who undertook to write, and did write an account of the above things, which were firmly believed by all; and which they took from the apostles, and first ministers of the Gospel, from their sermons and discourses, and from conversation with them; and which they committed to writing, partly to help their own memories, and partly for the benefit of others; in which, no doubt, they acted an upright part, though attended with weakness: wherefore, the evangelist does not censure them as false, wicked, and heretical, nor approve of them as divine and perfect for though they honestly meant, and designed well, yet there might be many things collected by them, which were impertinent, and not proper to be transmitted to posterity; and what might be wrote with great inaccuracy and deficiency, and in a style the Holy Ghost thought improper things of this kind should be delivered in: and therefore the evangelist, moved and inspired by the Spirit of God, set about the following work, and under the same influence completed it. The phrase, αναταξασθαι διηγησιν, &quot;to set forth in order a declaration&quot;, is as Dr. Lightfoot observes, out of the Talmud (h), agreeably to the Jewish way of speaking,
&quot;R. Chasdai said to one of the Rabbins, who was מסדר אגדתא, &quot;setting in order a declaration&quot; before him. &amp;c. or relating in order a story before him.
(h) T. Bab. Succa, fol. 53. 1.&quot;

Commentary by John Gill

&quot;Acts 1:1
The former treatise have I made,.... Meaning the Gospel written by him the Evangelist Luke, for from that he makes a transition to this, beginning here where he there left off; namely, at the ascension of Christ; see Luk_24:51.&quot;

Commentary by Peter Gusik

With further examination of the two books there are similarities in both style and choice of syntax which lend both books to a common authorship.

Hope this helps