Kent P. Jackson discusses the authorship of Isaiah.
Kent P. Jackson, "Isaiah in the Book of Mormon," in A Reason for Faith, ed. Laura Harris Hales (Provo: Religious Studies Center, BYU, 2016), 75-76
Latter-day Saints who accept the evidence from the Book of Mormon and believe that prophets can see beyond their own time should have no difficulty accepting the idea that the Isaiah chapters in the Book of Mormon were compiled before 600 BC. But this does not mean that all our questions have been answered. The great Old Testament scholar W. F. Albright pointed out that the prophetic books are not really books but rather “anthologies of oracles and sermons.” This description certainly fits the book of Isaiah.
Like the Bible itself, it is a collection. And, as with the Bible, the circumstances under which it was written and compiled are not clearly known. Did Isaiah record his prophecies himself, or did he dictate them to scribes? If they were dictated, was Isaiah responsible for their poetic structure, or were others? Did Isaiah gather and compile the revelations himself, or did others do it—even after his lifetime? Were Isaiah’s words edited or reworded by later scribes? Who is responsible for the final order of the prophecies in the book? And what is the book’s history in the century between Isaiah’s death and Nephi’s acquisition of the plates of brass?
The answers to these questions are not critical for our understanding of Isaiah’s message. But the questions show us that we cannot speak with certainty on many issues related to how and when the book of Isaiah became what it is today. What we do know is that Lehi and his sons had at least part of the book with them when they left Jerusalem and that Isaiah’s words are “great” because “all things that he spake have been and shall be, even according to the words which he spake.”